Sphere (EntityTopic, 15)

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{{Shape|Sphere|http://img457.imageshack.us/img457/787/sphere6jb.png|3|1, 0, 0|0|N/A|N/A|[[Line (object)|E]][[Circle|L]]L|3 (x,y,z)|N/A|N/A|N/A}}
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{{Shape|Sphere|http://img457.imageshack.us/img457/787/sphere6jb.png|3|1, 0, 0|0|N/A|N/A|[[Line (object)|E]][[Circle|L]]L|3 (x,y,z)|N/A|N/A|N/A|7}}
== Geometry ==
== Geometry ==
A '''sphere''' refers to the surface of a perfectly symmetrical [[realmic]] object.
A '''sphere''' refers to the surface of a perfectly symmetrical [[realmic]] object.
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=== Mapping ===
=== Mapping ===
When the surface of a sphere is mapped onto a [[rectangle]] {(-1,-1),(1,-1),(1,1),(-1,1)}, the surface will be horizontally stretched such that the further away from the equator of the sphere a point is, the more it is stretched. The top and bottom edges of the rectangle will converge into a single point. Leaving the top edge of the rectangle will continue entering from the top edge at the position (x±1,1). Similarly, leaving the bottom edge of the rectangle will continue entering from the bottom edge at the position (x±1,-1).
When the surface of a sphere is mapped onto a [[rectangle]] {(-1,-1),(1,-1),(1,1),(-1,1)}, the surface will be horizontally stretched such that the further away from the equator of the sphere a point is, the more it is stretched. The top and bottom edges of the rectangle will converge into a single point. Leaving the top edge of the rectangle will continue entering from the top edge at the position (x±1,1). Similarly, leaving the bottom edge of the rectangle will continue entering from the bottom edge at the position (x±1,-1).
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{{Rotope Nav|6|7|8|II'<br>Square pyramid|(III)<br>Sphere|I'I<br>Triangular prism}}
{{Polyhedra}}
{{Polyhedra}}
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{{Rotopes}}
 

Revision as of 11:58, 17 June 2007

Template:Shape

Geometry

A sphere refers to the surface of a perfectly symmetrical realmic object.

Equations

  • Assumption: Sphere is centered at the origin.
  • Variables:
r ⇒ radius of sphere
  • All points (x, y, z) that lie on the surface of a sphere will satisfy the following equation:
x2 + y2 + z2 = r2
total edge length = 0
surface area = 4πr2
volume = 4πr33-1
[!x,!y,!z] ⇒ circle of radius (rcos(πn/2))

Mapping

When the surface of a sphere is mapped onto a rectangle {(-1,-1),(1,-1),(1,1),(-1,1)}, the surface will be horizontally stretched such that the further away from the equator of the sphere a point is, the more it is stretched. The top and bottom edges of the rectangle will converge into a single point. Leaving the top edge of the rectangle will continue entering from the top edge at the position (x±1,1). Similarly, leaving the bottom edge of the rectangle will continue entering from the bottom edge at the position (x±1,-1). Template:Rotope Nav

Notable Trishapes
Regular: tetrahedroncubeoctahedrondodecahedronicosahedron
Direct truncates: tetrahedral truncatecubic truncateoctahedral truncatedodecahedral truncateicosahedral truncate
Mesotruncates: stauromesohedronstauroperihedronstauropantohedronrhodomesohedronrhodoperihedronrhodopantohedron
Snubs: snub staurohedronsnub rhodohedron
Curved: spheretoruscylinderconefrustumcrind