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Browse College Higher-Dimensional Geometry

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Name of Truncated Circle [12/29/2003]
What is the name of a rectangle with two rounded ends, like a circle with top and bottom evenly truncated? For example, an 'oval' racetrack with two straightaways on opposite sides, parallel to each other and both of the same length. I know that's not really an oval, but I'm not sure what else to call it!

n-Dimensional Cubes [09/06/2002]
Define #(n,A) to be the number of corners of an n-dimensional cube whose distance to 0 is greater than A. The limit (as n goes to infinity) #(n,A)/(2^n) = 1. How would I verify the limit statement?

Nearest Point on a Great Circle [05/27/2002]
Given points A, B, and C on the surface of a unit sphere, find the point P on the great circle defined by A and B that is nearest to C.

Obtaining Bearing from a Velocity Vector [08/01/2001]
I have the x, y, and z components of a velocity vector of an airplane, and must use this vector to calculate the bearing of the plane.

Packing 4 Spheres Into a Tetrahedron [09/03/99]
How can I find the dimensions of the smallest tetrahedron that can serve as a container for 4 spheres packed as snugly as possible?

Pappus' Centroid Theorem [06/05/2003]
What is the formula for the volume of a circular torus?

Parametric Form of Circle Equation [08/04/2003]
If I know the center and radius of a circle, and three points on the circle, can I find the parametric form of the circle equation in 3D space?

Parametric Sphere Formula [7/7/1996]
What is the parametric form of a circle in 3 space that passes through a particular 3 points?

Planar Approximation: Latitude and Longitude [04/18/2003]
I am trying to calculate the midpoint between cases of legionella and their nearest neighbor. How can I calculate the distance between two points below which they can be treated as if they were in a plane rather than on a sphere?

Planes Intersecting Space [11/24/2001]
Can we say that n planes divide space into at most 2^n regions?

Proof for Volume of a Segment of a Sphere [11/19/2001]
I am in need of assistance in proving the volume of a truncated spherical cap (or a segment of a sphere I think it is also called).

Radius of the Earth as an Ellipsoid [06/26/2000]
I have been given two equations to determine the radius of the earth for a given latitude, based on ellipsoid model WGS84. I get different answers...

The Reflection of a Line [04/21/1998]
How can I find a vector equation of the reflection of a line in three- dimensional space?

Resolving Pitch and Yaw [02/10/2003]
Is there an equation to find the resultant of pitch and yaw?

Roll of Paper [06/15/2003]
I am getting a paper rewinder that runs 6,000 ft a minute, and the roll is 50' high above the floor. How many miles and feet are there in this roll of paper and how long will it take to run?

The Second Octant [04/03/2002]
Where is the second octant? No one seems to know how to count the next octants after the first.

Sketching a Plane in Three Dimensional Space [11/15/2005]
I know that an equation like 2x + y + z = 3 represents a plane in three dimensions. How can I sketch that plane on the xyz axes? Also, how can I sketch a system of such equations to find the solution geometrically?

Small Section of a Sphere [01/10/2002]
Find the volume and the areas of each of the surfaces/faces of a small section of a sphere with "dimensions" delta r, delta theta, delta phi, in spherical coordinates.

Sphere Eversion [8/11/1996]
How do you mathematically turn a sphere inside out?

Spherical Geometry [02/10/2003]
How do I calculate whether two lines that lie on the surface of a sphere intersect, and if they do intersect, the point of that intersection?

Spherical Polygon Area [08/09/1999]
Could you please explain the formula for the area of a spherical polygon, and show how to determine the values of the thetas in it?

Spherical 'Rectangles' [05/13/2002]
How can I find the 'spherical rectangle' defined by a pair of corner points?

Steinmetz Solid [06/08/2003]
Two pipes (radius = r) cross each other normally. What is the common part volume?

Surface Area and Volume Derivative [10/30/2000]
For what 3D figures is the derivative of the volume formula equal to the formula for surface area? With respect to which variable would you need to differentiate?

Surface Area of an Ellipsoid [02/11/1997]
How do you calculate the surface area of an ellipsoid?

Surface Area of an n-dimensional Sphere [07/28/1997]
I was wondering how to calculate the surface area of a sphere in n dimensions.

Surface Area of a Sphere [10/03/1997]
How is the surface area of a sphere calculated, and why?

Surface Area of a Sphere [04/10/1998]
Can you derive the formula for the surface area of a sphere?

Surface Area of Cones and Pyramids [09/27/2003]
Can the method for finding the surface area of a pyramid be used as well to find the surface area of a cone?

Surface Area of Solid of Revolution [05/21/2001]
I tried to derive the formula for the surface area of a cone by taking the integral the circumference of the solid of revolution, but it didn't work. What did I do wrong? Can the formula be derived using this method?

Tesseract [04/25/2001]
Why does a tesseract contain eight cubes?

Tetrahedron Projected on a Plane [10/29/1996]
How do you project a regular tetrahedron perpendicularly onto a plane to get the maximum area shadow?

Tipped and Partially Filled Frustum [12/14/2003]
A vessel in a plant where I work is the frustum of a cone on its side. A liquid is contained in this section and pours out the end of the cone section, therefore the liquid only takes up a certain portion of the cone's volume. How can I compute the volume of the liquid?

Transformation between (x,y) and (longitude, latitude) [01/02/2002]
I have two questions on the transformation between (x,y) and (longitude, latitude).

Two Polygons in 3D Space [01/31/2003]
How do you find the shortest distance between two polygons in 3D space?

Understanding Fourth Dimension Figures [07/05/1998]
Can you help me figure out the equations for fourth dimension figures such as the tesseract and the hypertetrahedron?

Uniform Distribution of Random Points on a Sphere [07/14/2005]
Is there a method to pick random points on a sphere so that the points wind up uniformly distributed on the sphere? I keep getting a higher density of points near the poles.

Using Longitude and Latitude to Determine Distance [4/17/1995]
I've been looking for the equation for finding the distance between two cities, given the latitude and longitude of both cities.

Variable Volumes in an Oblate Spheroid [12/21/2002]
We need to know how much water is in the tank at any given time.

Vectors and the Volume of Parallelepipeds [03/08/2003]
Explain the derivation of the formula V = |a.(b x c)| (the volume of a parallelepiped is equal to the magnitude of the scalar triple product of the vectors that determine the parallelepiped; where a, b, and c are those vectors).

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