L’espace - 3e
Repérage sur une sphère
Exercice 1 : Lire latitude ou longitude d'une ville sur terre à partir des parallèles et méridiens
Sur le schéma ci-dessous, le point D symbolise la ville de La Mecque (Arabie saoudite).
{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240, "sub_curve": 64}, "size": {"width": 600, "height": 500}, "camera": {"starting_position": {"x": -534, "y": 221, "z": 76}}, "materials_options": {"surface": {"opacity": 1}, "solid_line": {"linewidth": 2}, "hidden_solid_line": {"linewidth": 2, "dashSize": 2, "gapSize": 2}}}, "data": {"Sphere_0": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "SpherePortion", "radius": 40, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "texture": "/static/img/earthmap1k.jpg", "bumpMap": "/static/img/earthbump1k.jpg", "bumpScale": 4, "specularMap": "/static/img/earthspec1k.jpg", "offset": {"x": 3.141592653589793}}, "plain_uv_mapping": true, "not_draw_inner": true}, "PlaneSurface_2": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16428120, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_1"}, "Arc_1": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_4": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -0.3490658503988659, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_3"}, "Arc_3": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_6": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.3490658503988659, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_5"}, "Arc_5": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_8": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -0.6981317007977318, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_7"}, "Arc_7": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_10": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.6981317007977318, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_9"}, "Arc_9": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_12": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -1.0471975511965976, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_11"}, "Arc_11": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_14": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 1.0471975511965976, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_13"}, "Arc_13": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_16": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -1.3962634015954636, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_15"}, "Arc_15": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_18": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 1.3962634015954636, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_17"}, "Arc_17": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_20": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -1.7453292519943295, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_19"}, "Arc_19": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_22": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 1.7453292519943295, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_21"}, "Arc_21": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_24": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -2.0943951023931953, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_23"}, "Arc_23": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_26": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 2.0943951023931953, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_25"}, "Arc_25": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_28": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -2.443460952792061, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_27"}, "Arc_27": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_30": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 2.443460952792061, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_29"}, "Arc_29": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_32": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": -2.792526803190927, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_31"}, "Arc_31": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_34": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 2.792526803190927, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_33"}, "Arc_33": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_36": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.141592653589793, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_35"}, "Arc_35": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "Segment3D_37": {"type": "Segment", "pt1": {"x": 0.0, "y": -45.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 45.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16711680, "linewidth": 4}}, "Point3D_N": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_N": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "text": "N", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_S": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_S": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "text": "S", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "PlaneSurface_39": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 255, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_38"}, "Circle_38": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 40.5, "y": 40.5}}}]}, "PlaneSurface_41": {"position": {"x": 0.0, "y": -13.680805733026748, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_40"}, "Circle_40": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 38.08770483143634, "y": 38.08770483143634}}}]}, "PlaneSurface_43": {"position": {"x": 0.0, "y": 13.680805733026748, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_42"}, "Circle_42": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 38.08770483143634, "y": 38.08770483143634}}}]}, "PlaneSurface_45": {"position": {"x": 0.0, "y": -25.71150438746157, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_44"}, "Circle_44": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 31.14177772475912, "y": 31.14177772475912}}}]}, "PlaneSurface_47": {"position": {"x": 0.0, "y": 25.71150438746157, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_46"}, "Circle_46": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 31.14177772475912, "y": 31.14177772475912}}}]}, "PlaneSurface_49": {"position": {"x": 0.0, "y": -34.64101615137754, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_48"}, "Circle_48": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 20.500000000000004, "y": 20.500000000000004}}}]}, "PlaneSurface_51": {"position": {"x": 0.0, "y": 34.64101615137754, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_50"}, "Circle_50": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 20.500000000000004, "y": 20.500000000000004}}}]}, "PlaneSurface_53": {"position": {"x": 0.0, "y": -39.39231012048832, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_52"}, "Circle_52": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 7.445927106677217, "y": 7.445927106677217}}}]}, "PlaneSurface_55": {"position": {"x": 0.0, "y": 39.39231012048832, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_54"}, "Circle_54": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 7.445927106677217, "y": 7.445927106677217}}}]}, "Point3D_D": {"type": "Point", "position": {"x": -28.79385241571817, "y": 13.680805733026748, "z": 24.160910942202147}, "point_render": {"color": 16711680, "size": 128}, "label_render": {"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -28.79385241571817, "y": 13.680805733026748, "z": 24.160910942202147}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}
On pourra faire pivoter la figure en cliquant dessus.
Légende
- Méridien d'origine Greenwich 0°
- Méridiens (espacés de 20°)
- Équateur 0°
- Parallèles (espacés de 20°)
- N : Pôle nord
- S : Pôle sud
- D : Ville de La Mecque
Donner la longitude de la ville de La Mecque (Arabie saoudite).
Exemples de format de réponses attendu : \(80°N\), \(60°S\), \(80°E\) ou \(20°O\)
Exemples de format de réponses attendu : \(80°N\), \(60°S\), \(80°E\) ou \(20°O\)
Exercice 2 : Calculer la longueur de l'équateur ou d'un méridien à partir du rayon/diamètre de la planète
Sur une planète de diamètre \( 5\:000\:\text{km} \), on définit l'équateur de la même manière que sur Terre.
Quelle est la longueur de l'équateur de cette planète ?
On arrondira le résultat au kilomètre près, suivi de l'unité qui convient.
On arrondira le résultat au kilomètre près, suivi de l'unité qui convient.
Exercice 3 : Lire latitude ou longitude d'une ville sur terre à partir des angles
Sur le schéma ci-dessous, le point D symbolise la ville de Tomakomai (Japon).
{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240, "sub_curve": 64}, "size": {"width": 600, "height": 500}, "camera": {"starting_position": {"x": -534, "y": 221, "z": 76}}, "materials_options": {"surface": {"opacity": 1}, "solid_line": {"linewidth": 2}, "hidden_solid_line": {"linewidth": 2, "dashSize": 2, "gapSize": 2}}}, "data": {"Sphere_0": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "SpherePortion", "radius": 40, "force_second_circle": true, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "texture": "/static/img/earthmap1k.jpg", "bumpMap": "/static/img/earthbump1k.jpg", "bumpScale": 4, "specularMap": "/static/img/earthspec1k.jpg", "offset": {"x": 3.141592653589793}}, "phiStart": 2.478367537831948, "phiEnd": 6.283185307179586, "plain_uv_mapping": true, "not_draw_inner": true}, "PlaneSurface_2": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 16685614, "depthTest": true, "phong": true, "opacity": 1}, "type": "PlaneSurface", "path": "Arc_1"}, "Arc_1": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -0.49086153142880334, "y": 39.99698807356578}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": 1.5585244804918115, "endAngle": 4.724660826687775}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_4": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 5.619960191421741, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 65280, "depthTest": true, "phong": true, "opacity": 0.2}, "type": "PlaneSurface", "path": "Arc_3"}, "Arc_3": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -39.99698807356578, "y": -0.4908615314287929}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": -1.5830681730979816, "endAngle": 0.01227184630308513}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_6": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 5.619960191421741, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 65280, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_5"}, "Arc_5": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 10.0, "startAngle": -1.5707963267948966, "endAngle": 0}}]}, "PlaneSurface_8": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 5.619960191421741, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 255, "depthTest": true, "phong": true, "opacity": 0.2}, "type": "PlaneSurface", "path": "Arc_7"}, "Arc_7": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -28.917178521767948, "y": -27.636873671604075}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": -0.01227184630308513, "endAngle": 0.7627634246606468}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_10": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 5.619960191421741, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 255, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_9"}, "Arc_9": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 25.666666666666664, "startAngle": 0, "endAngle": 0.7504915783575618}}]}, "PlaneSurface_12": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 5.619960191421741, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 16711680, "depthTest": true, "phong": true, "opacity": 0.2}, "type": "PlaneSurface", "path": "Arc_11"}, "Arc_11": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": 0.4908615314287905, "y": -39.99698807356578}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": 0.7382197320544767, "endAngle": 1.5830681730979816}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_14": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 5.619960191421741, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16711680, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_13"}, "Arc_13": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 24.333333333333336, "startAngle": 0.7504915783575618, "endAngle": 1.5707963267948966}}]}, "Segment3D_15": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": -45.0, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "Segment3D_16": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 35.460483912302486, "y": 0.0, "z": 27.70476638965463}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "Segment3D_17": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 25.934156170469464, "y": 30.68992620281243, "z": 20.26198345156829}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "PlaneSurface_19": {"position": {"x": 0.0, "y": 27.27993440249994, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 3.141592653589793, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 16776960, "opacity": 0.2, "depthTest": true}, "type": "PlaneSurface", "path": "Arc_18"}, "Arc_18": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": 23.05258326263953, "y": -18.010651956949584}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 29.254148064766817, "startAngle": 0, "endAngle": 2.478367537831948}}, {"lineTo": {"x": 0, "y": 0}}]}, "Segment3D_20": {"type": "Segment", "pt1": {"x": 0.0, "y": 27.27993440249994, "z": 0.0}, "pt2": {"x": -34.25414806476682, "y": 27.27993440249994, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "Segment3D_21": {"type": "Segment", "pt1": {"x": 0.0, "y": 27.27993440249994, "z": 0.0}, "pt2": {"x": 26.992637030673134, "y": 27.27993440249994, "z": 21.088959333577886}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "PlaneSurface_23": {"position": {"x": 0.0, "y": 27.27993440249994, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 3.141592653589793, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16776960, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_22"}, "Arc_22": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 14.627074032383408, "startAngle": 0, "endAngle": 2.478367537831948}}]}, "Point3D_B": {"type": "Point", "position": {"x": -1.6704142173485658e-15, "y": 27.27993440249994, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_B": {"type": "Label", "position": {"x": -1.6704142173485658e-15, "y": 27.27993440249994, "z": 0.0}, "text": "B", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "PlaneSurface_25": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 6949000, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_24"}, "Arc_24": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_27": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 2.478367537831948, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_26"}, "Arc_26": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "Segment3D_28": {"type": "Segment", "pt1": {"x": 0.0, "y": -45.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 45.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16711680, "linewidth": 4}}, "Point3D_N": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_N": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "text": "N", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_S": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_S": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "text": "S", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "PlaneSurface_30": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 255, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_29"}, "Circle_29": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 40.5, "y": 40.5}}}]}, "PlaneSurface_32": {"position": {"x": 0.0, "y": 27.27993440249994, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_31"}, "Circle_31": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 29.754148064766817, "y": 29.754148064766817}}}]}, "Point3D_C": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_C": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "text": "C", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_E": {"type": "Point", "position": {"x": 31.520430144268875, "y": 0.0, "z": 24.626459013026334}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": 31.520430144268875, "y": 0.0, "z": 24.626459013026334}, "text": "E", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_A": {"type": "Point", "position": {"x": -29.254148064766817, "y": 27.27993440249994, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -29.254148064766817, "y": 27.27993440249994, "z": 0.0}, "text": "A", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_D": {"type": "Point", "position": {"x": 23.052583262639523, "y": 27.27993440249994, "z": 18.010651956949594}, "point_render": {"color": 16711680, "size": 128}, "label_render": {"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": 23.052583262639523, "y": 27.27993440249994, "z": 18.010651956949594}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}
On pourra faire pivoter la figure en cliquant dessus.
Légende
- Méridien d'origine Greenwich 0°
- Méridien du point D
- Équateur 0°
- Parallèle du point D
- N : Pôle nord
- S : Pôle sud
- C : Centre de la terre
- D : Ville de Tomakomai
Sachant que \( \widehat{DCN} = 47 ° \), \( \widehat{ABD} = 142 ° \), \( \widehat{ECD} = 43 ° \) et \( \widehat{SCE} = 90 ° \), donner la longitude de la ville de Tomakomai (Japon).
Exemples de format de réponses attendu : \(80°N\), \(60°S\), \(80°E\) ou \(20°O\)
Exemples de format de réponses attendu : \(80°N\), \(60°S\), \(80°E\) ou \(20°O\)
Exercice 4 : Lire les coordonnées géographiques d'une ville sur terre à partir des angles
Sur le schéma ci-dessous, le point D symbolise la ville de Brateyevo (Russie).
{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240, "sub_curve": 64}, "size": {"width": 600, "height": 500}, "camera": {"starting_position": {"x": -534, "y": 221, "z": 76}}, "materials_options": {"surface": {"opacity": 1}, "solid_line": {"linewidth": 2}, "hidden_solid_line": {"linewidth": 2, "dashSize": 2, "gapSize": 2}}}, "data": {"Sphere_0": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "SpherePortion", "radius": 40, "force_second_circle": true, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "texture": "/static/img/earthmap1k.jpg", "bumpMap": "/static/img/earthbump1k.jpg", "bumpScale": 4, "specularMap": "/static/img/earthspec1k.jpg", "offset": {"x": 3.141592653589793}}, "phiStart": 0.6632251157578453, "phiEnd": 6.283185307179586, "plain_uv_mapping": true, "not_draw_inner": true}, "PlaneSurface_2": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 16685614, "depthTest": true, "phong": true, "opacity": 1}, "type": "PlaneSurface", "path": "Arc_1"}, "Arc_1": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -0.49086153142880334, "y": 39.99698807356578}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": 1.5585244804918115, "endAngle": 4.724660826687775}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_4": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.8048177693476384, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 65280, "depthTest": true, "phong": true, "opacity": 0.2}, "type": "PlaneSurface", "path": "Arc_3"}, "Arc_3": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -39.99698807356578, "y": -0.4908615314287929}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": -1.5830681730979816, "endAngle": 0.01227184630308513}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_6": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.8048177693476384, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 65280, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_5"}, "Arc_5": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 10.0, "startAngle": -1.5707963267948966, "endAngle": 0}}]}, "PlaneSurface_8": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.8048177693476384, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 255, "depthTest": true, "phong": true, "opacity": 0.2}, "type": "PlaneSurface", "path": "Arc_7"}, "Arc_7": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -21.959089238465705, "y": -33.43349218698369}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": -0.01227184630308513, "endAngle": 0.9896562274199097}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_10": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.8048177693476384, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 255, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_9"}, "Arc_9": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 21.333333333333332, "startAngle": 0, "endAngle": 0.9773843811168246}}]}, "PlaneSurface_12": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.8048177693476384, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 16711680, "depthTest": true, "phong": true, "opacity": 0.2}, "type": "PlaneSurface", "path": "Arc_11"}, "Arc_11": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": 0.4908615314287905, "y": -39.99698807356578}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.0, "startAngle": 0.9651125348137395, "endAngle": 1.5830681730979816}}, {"lineTo": {"x": 0, "y": 0}}]}, "PlaneSurface_14": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 3.8048177693476384, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16711680, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_13"}, "Arc_13": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 28.666666666666668, "startAngle": 0.9773843811168246, "endAngle": 1.5707963267948966}}]}, "Segment3D_15": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": -45.0, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "Segment3D_16": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": -35.460483912302486, "y": 0.0, "z": 27.704766389654623}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "Segment3D_17": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": -19.829250957398134, "y": 37.30669076497688, "z": 15.492308757409727}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "PlaneSurface_19": {"position": {"x": 0.0, "y": 33.16150290220167, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 3.141592653589793, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "color": 16776960, "opacity": 0.2, "depthTest": true}, "type": "PlaneSurface", "path": "Arc_18"}, "Arc_18": {"type": "Path2D", "path": [{"moveTo": {"x": 0, "y": 0}}, {"lineTo": {"x": -17.626000851020564, "y": -13.770941117697534}}, {"absArc": {"center": {"x": 0, "y": 0}, "radius": 22.36771613882987, "startAngle": 0, "endAngle": 0.6632251157578453}}, {"lineTo": {"x": 0, "y": 0}}]}, "Segment3D_20": {"type": "Segment", "pt1": {"x": 0.0, "y": 33.16150290220167, "z": 0.0}, "pt2": {"x": -27.36771613882987, "y": 33.16150290220167, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "Segment3D_21": {"type": "Segment", "pt1": {"x": 0.0, "y": 33.16150290220167, "z": 0.0}, "pt2": {"x": -21.56605461905417, "y": 33.16150290220167, "z": 16.849248494325828}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 0, "linewidth": 2}}, "PlaneSurface_23": {"position": {"x": 0.0, "y": 33.16150290220167, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 3.141592653589793, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16776960, "linewidth": 2}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_22"}, "Arc_22": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 11.183858069414935, "startAngle": 0, "endAngle": 0.6632251157578453}}]}, "Point3D_B": {"type": "Point", "position": {"x": -2.030556419204847e-15, "y": 33.16150290220167, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_B": {"type": "Label", "position": {"x": -2.030556419204847e-15, "y": 33.16150290220167, "z": 0.0}, "text": "B", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "PlaneSurface_25": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 6949000, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_24"}, "Arc_24": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "PlaneSurface_27": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.6632251157578453, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 11294970, "linewidth": 4}, "surface_render": {"visible": false}, "opened": true, "type": "PlaneSurface", "path": "Arc_26"}, "Arc_26": {"type": "Path2D", "path": [{"absArc": {"center": {"x": 0, "y": 0}, "radius": 40.5, "startAngle": 1.5707963267948966, "endAngle": 4.71238898038469}}]}, "Segment3D_28": {"type": "Segment", "pt1": {"x": 0.0, "y": -45.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 45.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16711680, "linewidth": 4}}, "Point3D_N": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_N": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "text": "N", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_S": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_S": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "text": "S", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "PlaneSurface_30": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 255, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_29"}, "Circle_29": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 40.5, "y": 40.5}}}]}, "PlaneSurface_32": {"position": {"x": 0.0, "y": 33.16150290220167, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_31"}, "Circle_31": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 22.86771613882987, "y": 22.86771613882987}}}]}, "Point3D_C": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_C": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "text": "C", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_E": {"type": "Point", "position": {"x": -31.520430144268875, "y": 0.0, "z": 24.62645901302633}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": -31.520430144268875, "y": 0.0, "z": 24.62645901302633}, "text": "E", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_A": {"type": "Point", "position": {"x": -22.36771613882987, "y": 33.16150290220167, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -22.36771613882987, "y": 33.16150290220167, "z": 0.0}, "text": "A", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_D": {"type": "Point", "position": {"x": -17.62600085102056, "y": 33.16150290220167, "z": 13.770941117697534}, "point_render": {"color": 16711680, "size": 128}, "label_render": {"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -17.62600085102056, "y": 33.16150290220167, "z": 13.770941117697534}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}
On pourra faire pivoter la figure en cliquant dessus.
Légende
- Méridien d'origine Greenwich 0°
- Méridien du point D
- Équateur 0°
- Parallèle du point D
- N : Pôle nord
- S : Pôle sud
- C : Centre de la terre
- D : Ville de Brateyevo
Sachant que \( \widehat{ECD} = 56 ° \), \( \widehat{SCE} = 90 ° \), \( \widehat{DCN} = 34 ° \) et \( \widehat{ABD} = 38 ° \), donner les coordonnées géographique de la ville de Brateyevo (Russie).
Par exemple:
Par exemple:
- Pour la ville de Bonn (Allemagne) on donnera une réponse de la forme \(\left(51° N; 7° E\right)\)
- Pour la ville d'Eloy Alfaro (Équateur) on donnera une réponse de la forme \(\left(2° S; 80° O\right)\)
Exercice 5 : Calculer le rayon d'un parallèle
On assimile la Terre à une sphère de rayon \( 6\:400\:\text{km} \).
{"options": {"render": {"sprite_scale": 0.02, "label_fontsize": 240, "sub_curve": 64}, "size": {"width": 600, "height": 500}, "camera": {"starting_position": {"x": -534, "y": 221, "z": 76}}, "materials_options": {"surface": {"opacity": 1}, "solid_line": {"linewidth": 2}, "hidden_solid_line": {"linewidth": 2, "dashSize": 2, "gapSize": 2}}}, "data": {"Sphere_0": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 0.0, "y": 0.0, "z": 0.0, "order": "XYZ"}, "type": "SpherePortion", "radius": 40, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": false}, "surface_render": {"visible": true, "texture": "/static/img/earthmap1k.jpg", "bumpMap": "/static/img/earthbump1k.jpg", "bumpScale": 4, "specularMap": "/static/img/earthspec1k.jpg", "offset": {"x": 3.141592653589793}}, "plain_uv_mapping": true, "not_draw_inner": true}, "Segment3D_1": {"type": "Segment", "pt1": {"x": 0.0, "y": -45.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 45.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 16711680, "linewidth": 4}}, "Point3D_N": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_N": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": 40.0, "z": 0.0}, "text": "N", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_S": {"type": "Point", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_S": {"type": "Label", "position": {"x": -2.4492935982947065e-15, "y": -40.0, "z": 0.0}, "text": "S", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "PlaneSurface_3": {"position": {"x": 0.0, "y": 0.0, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 255, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_2"}, "Circle_2": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 40.5, "y": 40.5}}}]}, "PlaneSurface_5": {"position": {"x": 0.0, "y": 20.601522996402167, "z": 0.0}, "rotation": {"x": 1.5707963267948966, "y": 0.0, "z": 0.0, "order": "XYZ"}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"visible": true, "color": 8487415, "linewidth": 4}, "surface_render": {"visible": false}, "type": "PlaneSurface", "path": "Circle_4"}, "Circle_4": {"type": "Path2D", "path": [{"absEllipse": {"center": {"x": 0.0, "y": 0.0}, "startAngle": 0, "endAngle": 6.28318530718, "radius": {"x": 34.786692028084495, "y": 34.786692028084495}}}]}, "Point3D_D": {"type": "Point", "position": {"x": -10.024458597554533, "y": 20.601522996402167, "z": 32.788526652697094}, "point_render": {"color": 16711680, "size": 128}, "label_render": {"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -10.024458597554533, "y": 20.601522996402167, "z": 32.788526652697094}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}
On pourra faire pivoter la figure en cliquant dessus.
Légende
- Équateur 0°
- Parallèle passant par D
- N : Pôle nord
- S : Pôle sud
- D : Ville de Gojra
Déterminer le rayon du parallèle passant à Gojra de coordonnées (31° N, 73° E).
On arrondira donnera un résultat arrondi au \( km \).
On arrondira donnera un résultat arrondi au \( km \).