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Assignment 1 (Due on Jan 23, 1997)

Consider a 3D line parameterized by:

where is an arbitrary point, , on the line and is an unit vector, , (also known as the direction cosine) along the line. Let and be any four points along the line defined by

Assume that the focal length of the camera is 1. Answer the following questions.

  1. Let us denote the distance between two 3D points and as . Derive expressions for (i) , (ii) , (iii) , and (iv) .

  2. Let and be the 2D images of , and , respectively, under perspective projection. Write down the expressions for the 2D image coordinates.

  3. Derive an expression for the distance between and (on the image plane). Let us denote this distance by

  4. Write expressions for the distances (i) , (ii) , and (iii)

  5. What is the value for ? This ratio is referred to as the cross ratio.

Note that the cross ratio is independent of the reference frame, the point through which the line goes, and the direction cosine of the line. It depends only on the 3D distances between the points. Thus, the cross-ratio is invariant under perspective projection.

Sudeep Sarkar
Wed Feb 12 17:37:05 EST 1997