Linear Transformation of a Sphere

Michael Tritle

09/09/2019

MATH 401: 3D Printing

Assignment 0

Consider two vectors u,v in V and a scalar q in F, where V is a vector space, and F is a field. A transformation is called linear if it satisfies two properties.

Intuitively and geometrically, we tend to think of linear transformations as rotations and shifts while keeping grid lines intact. If we were to consider a Cartesian Plane of real numbers, we see it like this:

u

T(u)

Now let’s consider a sphere that lives in 3D real space. There are a few different ways to represent a sphere in 3 dimensions. We can consider the set of all points that is equidistant from the origin. Using the distance formula in 3 dimensions we achieve

where k is an arbitrary real valued constant. Let to apply a linear transformation to a sphere, it would be easier to work with this in vector form. Using spherical coordinates, it’s a straight forward procedure:

 We’d now like to find out to find a basis for R3 that will represent the boundaries of our sphere of radius 1. Letting theta = pi/2 and phi = 0 results in (1,0,0), letting theta = pi/2 and phi = pi/2 results (0,1,0) in and letting and letting theta = 0 and phi = pi/2 results in (0,0,1).

Let’s now define a linear transformation that scales the boundaries of our sphere to and. Such a transformation can be represented by the matrix 

Left multiplication on A will transform our vector representation of a sphere as wanted;

If we write this as an implicit equation, we find that

                        

This is an equation for an ellipsoid! So the linear transformation we defined mapped the coordinates of our sphere to the coordinates of an ellipsoid. It might be an interesting exercise to see if all non trivial linear transformations from  on an sphere are mapped to an ellipsoid.

Creating a 3D printable representation of this ellipsoid can be made in openSCAD by scaling a sphere by it’s  components. The code used for the print is:

scale([3.0,2.0,1.0]) sphere(r=5.0);

Below are some pictures of the ellipsoid as a 3D model, and as a 3D print.