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[Texts]## 51M05: General Euclidean geometry |

We use this category to hold files concerning non-planar Euclidean geometry topics.

The files on this page are more like samples of the techniques one may use for 3D problems (or n-dimensional: much of what is here is really independent of the number of dimensions.)

The actions of the point groups among the crystallographic groups are the basis for the construction of the Platonic solids and the regular divisions of the sphere in R^3. For more information, consult the polyhedra and spheres pages.

Parent field: 51M - Real and Complex Geometry

For computational geometry see 68U05: Computer Graphics

Pointer to Mesa, a 3-D graphics library (similar to OpenGL).

- A FAQ: Shortest distance between two lines in 3-dimensional space.
- A sample post showing the use of vector methods for solving 3D problems involving straight objects (lines, planes, vectors, angles, etc.)
- How is geometry different in four-dimensional space?
- Pappus's Theorem: volume of a solid of revolution
- Pappus' theorems on centroids
- Kiss precise -- the radii of mutually tangent spheres in R^n
- A sample post showing the use of calculus techniques (finding the surface area of a baseball using Stokes' theorem).
- Computing the area of a triangle in 3-dimensional space
- Solid angle subtended by a polygon
- Some posts using group theory to analyze symmetry in 3D (the "space groups" -- useful for classifying regular solids too.)
- Sometimes what you need is really linear algebra -- in this case, describing rotations in 3D.
- Some background on the convex-hull problem (finding the points which form the "outside" of a set of points in space).
- An interesting problem in Euclidean geometry: show that a map which sends spheres to spheres must be an isometry.
- How many lines pass through four given lines in R^3 (two; generalize?) This is Enumerative Geometry (use the Schubert calculus).
- How many cylinders pass through five given points?
- Suppose L is a set in space such that all lines through 2 points in L passes through a third. Then L is collinear.
- Subdivisions of the sphere corresponding to the actions of the dihedral group.
- Finding a fundamental domain for the action of a group of symmetries of a sphere

Last modified 2000/01/14 by Dave Rusin. Mail: