Circles on a Sphere

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2021-12-30 22:06

In collections Curves and Spirals on Spheres

This example demonstrates circles on a sphere. It involves key concepts in spherical geometry, and uses the Sequence command in GeoGebra to dynamically generate a series of circles. Construction highlights include using the Sequence command to create a series of circles with varying radii and diameters. Suitable for high school and above students or teachers for teaching and research.

GeoGebra Commands & Parameters

Sequence(CurveCartesian[((r * sin(u)) * cos(v)), ((r * sin(u)) * sin(v)), (r * cos(u)), u, 0, (2 * π)], v, 0, (2 * π), π / d) → list1, {(((3.2 * sin(u)) * cos(0)), ((3.2 * sin(u)) * sin(0)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(0.39269908169872414)), ((3.2 * sin(u)) * sin(0.39269908169872414)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(0.7853981633974483)), ((3.2 * sin(u)) * sin(0.7853981633974483)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(1.1780972450961724)), ((3.2 * sin(u)) * sin(1.1780972450961724)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(1.5707963267948966)), ((3.2 * sin(u)) * sin(1.5707963267948966)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(1.9634954084936207)), ((3.2 * sin(u)) * sin(1.9634954084936207)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(2.356194490192345)), ((3.2 * sin(u)) * sin(2.356194490192345)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(2.748893571891069)), ((3.2 * sin(u)) * sin(2.748893571891069)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(3.141592653589793)), ((3.2 * sin(u)) * sin(3.141592653589793)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(3.5342917352885173)), ((3.2 * sin(u)) * sin(3.5342917352885173)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(3.9269908169872414)), ((3.2 * sin(u)) * sin(3.9269908169872414)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(4.319689898685965)), ((3.2 * sin(u)) * sin(4.319689898685965)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(4.71238898038469)), ((3.2 * sin(u)) * sin(4.71238898038469)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(5.105088062083414)), ((3.2 * sin(u)) * sin(5.105088062083414)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(5.497787143782139)), ((3.2 * sin(u)) * sin(5.497787143782139)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(5.890486225480863)), ((3.2 * sin(u)) * sin(5.890486225480863)), (3.2 * cos(u))), (((3.2 * sin(u)) * cos(6.283185307179588)), ((3.2 * sin(u)) * sin(6.283185307179588)), (3.2 * cos(u)))}
Sequence(CurveCartesian[((r * sin(u)) * cos(v)), ((r * sin(u)) * sin(v)), (r * cos(u)), v, 0, (2 * π)], u, (-π), π, π / d) → list2, {(((3.2 * sin(-3.141592653589793)) * cos(v)), ((3.2 * sin(-3.141592653589793)) * sin(v)), (3.2 * cos(-3.141592653589793))), (((3.2 * sin(-2.748893571891069)) * cos(v)), ((3.2 * sin(-2.748893571891069)) * sin(v)), (3.2 * cos(-2.748893571891069))), (((3.2 * sin(-2.356194490192345)) * cos(v)), ((3.2 * sin(-2.356194490192345)) * sin(v)), (3.2 * cos(-2.356194490192345))), (((3.2 * sin(-1.9634954084936207)) * cos(v)), ((3.2 * sin(-1.9634954084936207)) * sin(v)), (3.2 * cos(-1.9634954084936207))), (((3.2 * sin(-1.5707963267948966)) * cos(v)), ((3.2 * sin(-1.5707963267948966)) * sin(v)), (3.2 * cos(-1.5707963267948966))), (((3.2 * sin(-1.1780972450961724)) * cos(v)), ((3.2 * sin(-1.1780972450961724)) * sin(v)), (3.2 * cos(-1.1780972450961724))), (((3.2 * sin(-0.7853981633974483)) * cos(v)), ((3.2 * sin(-0.7853981633974483)) * sin(v)), (3.2 * cos(-0.7853981633974483))), (((3.2 * sin(-0.39269908169872414)) * cos(v)), ((3.2 * sin(-0.39269908169872414)) * sin(v)), (3.2 * cos(-0.39269908169872414))), (((3.2 * sin(0)) * cos(v)), ((3.2 * sin(0)) * sin(v)), (3.2 * cos(0))), (((3.2 * sin(0.39269908169872414)) * cos(v)), ((3.2 * sin(0.39269908169872414)) * sin(v)), (3.2 * cos(0.39269908169872414))), (((3.2 * sin(0.7853981633974483)) * cos(v)), ((3.2 * sin(0.7853981633974483)) * sin(v)), (3.2 * cos(0.7853981633974483))), (((3.2 * sin(1.1780972450961724)) * cos(v)), ((3.2 * sin(1.1780972450961724)) * sin(v)), (3.2 * cos(1.1780972450961724))), (((3.2 * sin(1.5707963267948966)) * cos(v)), ((3.2 * sin(1.5707963267948966)) * sin(v)), (3.2 * cos(1.5707963267948966))), (((3.2 * sin(1.9634954084936207)) * cos(v)), ((3.2 * sin(1.9634954084936207)) * sin(v)), (3.2 * cos(1.9634954084936207))), (((3.2 * sin(2.356194490192345)) * cos(v)), ((3.2 * sin(2.356194490192345)) * sin(v)), (3.2 * cos(2.356194490192345))), (((3.2 * sin(2.748893571891069)) * cos(v)), ((3.2 * sin(2.748893571891069)) * sin(v)), (3.2 * cos(2.748893571891069))), (((3.2 * sin(3.141592653589793)) * cos(v)), ((3.2 * sin(3.141592653589793)) * sin(v)), (3.2 * cos(3.141592653589793)))}

AI Drawing Prompt Reference

Create a 3D sphere of radius 3.2. On the sphere, generate two families of circles from spherical parameterizations. For the first family, form latitudinal circles by varying u from 0 to 2π in d equal steps (e.g., d=16); for each chosen u, trace the circle using points with v running from 0 to 2π. For the second family, repeat with u varying from −π to π in the same d steps, and again trace each resulting circle with v from 0 to 2π. The two interlaced sets of circles should lie on the sphere and collectively illustrate circles on a spherical surface.

The prompt above was compiled from the original drawing commands and is for reference only.