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Browse College Higher-Dimensional Geometry

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Changing Angle of a Tank [06/11/2003]
Points A and B represent pressure sensors in fixed positions on the base of a round tank. The chord through CD represents the water level in the tank. Lines a and b are the heights of water registered by each sensor...

3D Geometry [11/17/1997]
You can draw a line of minimum distance between and perpendicular to two lines in 3space. I know how to get the distance and direction of this line, but I want to locate the line in 3space so that I can find its midpoint.

3-Dimensional Rotation Space [05/18/2009]
Consider a closed loop representing a rotation of 2pi in RP^3. Can you show that one cannot continuously deform this loop to a point?

3D Projection Onto a 2D Plane [7/9/1996]
I've got a geological problem: I want to take the points of a 3D (but almost planar) ore body, fit them to a plane, do some planar analysis on those points, and then recover the original points.

Angle Between Two Points on the Globe [7/17/1996]
Given their longitude and latitude, how can you determine the angle in radians between two cities?

Area of a Latitude-Longitude Rectangle [07/31/2003]
Given the latitudes and longitudes of four points on the Earth's surface, how do you calculate the surface area enclosed by the four points?

Bearing Calculation [09/01/1997]
Given two cities at geographic coordinates (xA,yA) and (xB,yB), is there a formula to calculate the bearing from city A to city B?

Building a Cone [01/28/2002]
I am trying to draw cone (frustum) with a larger radius size.

Calculating How Much Paper Is on a Roll [04/24/2006]
How many feet of paper are on a roll of paper with a diameter of 48" if the core has a diameter of 4" and the paper is 0.014" thick?

Calculating the True Bearing Between Two Points [9/20/1995]
Given two points on the earth's surface - their grid coordinates, and their lat-long coordinates, and knowing how to get from the grid coordinates to a grid bearing - how can I calculate the true bearing between the two points?

Calculating the Volume of a Sphere in N-Dimensional Space [9/5/1995]
How do I calculate the volume of a sphere in 4-dimension or, more generally, in n-dimension space?

Circles on the Surface of a Sphere [08/11/1999]
How can I convert the equation of a circle on a unit sphere in the (X,Y,Z) coordinate system into a function that is defined in terms of spherical coordinates theta and phi?

Circumference of a Tube [10/11/2001]
What effect does the thickness of a tube have on the circumference when it's formed into a circle?

Compound Angles [01/14/2001]
A rail does not go into a post horizontally, but has a rise angle of 30 degrees and a side angle of 39 degrees. The hole to be routed needs to be a parallelogram instead of a square...

Computing Altitude from an Earth-Centered Position [01/04/2002]
I'm looking for a closed form algorithm that provides altitude above the reference (WGS-84) ellipsoid given an Earth-centered position.

Construction of a Regular Heptadecagon [12/27/2009]
Gauss derived a finite algebraic expression for sin(pi/17) which led to an algorithm for the construction of the regular 17-gon. Can you help me understand the derivation of his expression?

Cube in a Cone [12/17/2002]
I need to find the length of each side of the cube.

Cubic Footage of a Tapered Log [08/07/2002]
Is there an equation that will calculate the cubic footage of a log given the diameters of each end and the length?

Cylinder of Arbitrary Axis [9/2/1996]
What is the equation of a cylinder about an arbitrary axis...?

Dead Reckoning [08/12/2003]
I am looking to do a form of 'dead reckoning' using a fixed latitude/longitude position, velocity components for north and west, and a time delay to compute an extrapolated latitude/longitude position.

Defining an Ellipse [07/16/2003]
A right cylinder of radius 'r' (a lift duct) intercepts a plane (the deck of a hovercraft) at an angle of 30 degrees to the vertical. How may the resulting ellipse be scribed? - i.e., how do I mark the hole to be cut in the deck?

Definitions of Edge and Face in 2D and 3D [10/10/2008]
What is the 'official' definition of 'edge'? Specifically, is an edge restricted to the intersection of two non-coplanar faces or do two- dimensional shapes have edges? I'm also curious about a definition of 'face'. How many faces does a two-dimensional shape have?

Derivation of Formula for Surface Area of Torus [05/05/2005]
I saw the formula for the surface area of a torus on your web site, and I'm wondering if you can show me how that formula was derived?

Derivation of Geometric Formulas [5/29/1996]
What are the formulas for the surface area, total surface area, and volume of a sphere, and volume of a pyramid and cone?

Deriving the Area of a Sphere [10/21/2003]
I know the area of a sphere is 4phi(r^2), but I'm wondering how to derive that formula. I know it should be done in cylindrical coordinates, and I'm thinking that the arc of a circle is defined as rd(theta) and it's multiplied with rd(phi) to get (r^2)d(theta)d(phi). Could you please help explain this?

Desargues' Theorem [07/03/1998]
Why is Desargues' two triangle theorem easy to prove in three dimensions but impossible in two dimensions?

Distance Between Two Points on the Earth [6/21/1996]
My latitude and longitude are in the form 40.266934, -74.204930 respectively, with negatives for South and West. How do I calculate the distance between them?

Distance from Circle to Line in 3-D [8/9/1996]
What is the analytic formula for the nearest distance from a circle to a line or line segment in 3-space?

Ellipsoid and Plane Intersection Equation [04/20/2000]
If an ellipsoid has half-axes a, b, and c, and the plane is normal to the vector [i,j,k] and also passes through the point (i,j,k), what are the half-axes and orientation of the ellipse of intersection?

Enneper's Minimal Surface [10/22/2001]
What is an Enneper Minimal Surface?

Equation of a Line in 3-D Space [03/18/2004]
Given two points in 3-D space, such as A(x1,y1,z1) and B(x2,y2,z2), what would be the equation of the line that connects those points? I know that in the 2-D plane the equation of a line in slope-intercept form is y = mx + b. Is there something similar in 3-D?

Equation of a Line in Three or More Dimensions [05/18/2000]
Can the equation y = mx + b be used to define a line in three dimensions? What about four or more dimensions?

Equation of an Ellipse in 3-Space [07/02/2003]
I am looking for the equation of an ellipse in 3-dimensional space. It can be a parametric formulation (e.g., x(t), y(t), z(t)) or a more canonical form (e.g., the 3D analog to the 2D form ((X*X)/a)+((Y*Y)/ b)=1).

Equation of a Plane [05/09/1997]
How do you find the equation of a plane when any three points on it are given or when a point and the equation of the normal are given?

Equation of a Sphere [09/17/1997]
We are trying to find an equation that would solve for the center point and radius of a sphere given three points in 3D space.

Finding a Best-Fit Regression Plane [12/15/2005]
I have a height map of a terrain where the x and y values are fixed. I can calculate best-fit slopes in the x and y directions, but I can't figure out how to combine them into a best-fit regression plane.

Finding an Ellipse in 3-D Space [06/14/2005]
How can I fit an ellipse to a set of 3-dimensional data points?

Finding Dimensions of Elliptical Shadow Cast on Plane [01/19/2005]
A sphere is positioned above a plane and light is shown on it from one side, resulting in an elliptical shadow on the plane. How can you determine the lengths of the axes of the resulting ellipse?

Finding Points on the Earth [06/08/2001]
Find the point that has latitude and longitude five miles north of a given point, and the other three points to the south, east, and west.

Finding the Angle of Solar Collectors on a Sloped Roof [08/30/2008]
I have solar collectors on my roof. They are mounted so that the base of each panel runs up the slope of the roof, and the panels themselves are mounted at an angle. I'd like to know how to determine the various angles created by this situation.

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