Great circles through the two poles are called lines or meridians of longitude and connecting is axis rotation. All these other surfaces would have boundaries and sphere is only that lacks boundary with constant positive Gaussian curvature. The pseudosphere is an example of surface with constant negative Gaussian curvature. The centers of all ordinary spheres pencil lie on line passing through and perpendicular to radical plane

Read More →Moreover a sphere orthogonal to any two spheres of pencil is all them and its center lies in radical plane . Numerous other closed convex surfaces have constant width for example Meissner body. Therefore sphere is closed

Read More →The shortest distance along surface between two distinct nonantipodal points sphere length of smaller arcs unique great circle that includes . Consequently a sphere is uniquely determined by that passes through circle and point not the plane of . Circular cylinders are the only surfaces with twoparameter families of rigid motions and revolution helicoids oneparameter family.

Read More →Borwein. Many theorems from classical geometry hold true for spherical well but not all do because sphere fails to satisfy some of postulates including parallel . displaystyle delta V approx pi cdot . This means that every point on the sphere will be umbilical . displaystyle pi r

Read More →Alternatively this formula found using spherical coordinates with volume element displaystyle dV theta dr varphi so . Topology edit In an nsphere is defined as space homeomorphic the boundary of ball thus Euclidean but perhaps lacking its metric. Therefore a threeparameter family of rotations exists such that each transforms sphere onto itself this group SO

Read More →C function use strict var k G. The hemisphere conjectured to be optimal least area isometric filling of Riemannian circle. Also any two similar spherical triangles are congruent. The surface area of unit sphere is displaystyle frac pi Gamma left right where z Euler function

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The sphere is only imbedded surface that lacks boundary or singularities with constant positive mean curvature. If the spheres intersect an imaginary circle all of pencil also pass through this but ordinary they are disjoint have no real points common