CS262 Homework 1: Interpolation

Due Thursday, October 17, Turn in and Present in Class

Objectives:

This assignment is to clarify the results of different methods of interpolating rotations in three dimensions. You have to turn in your solutions on paper, but you can program the computer to help if you like. In any case, your solutions should be clear and easy to follow. Show the steps.

Assignment:

The base problem is to rotate between two coordinate frames. Image a card that is white on the side perpendicular to the Y-axis and black on the other. It is attached by a stock to the world space origin. In the default position, the card and stick lie along the positive Z axis, and the white side is pointing in the direction of the Y axis.

Example 1: In orientation R1, the stick is lying along the X axis and the white side is still pointing in the direction of the Y axis. In orientation R2, the stick is lying along the Y axis, and the white side is pointing in the direction of the negative Z axis.

Example 2: In orientation R1, the stick is lying along the X axis and the white side is still pointing in the direction of the Y axis. In orientation R2, the stick is lying along the Y axis, and the white side is pointing in the direction of the X axis.

For these two examples, find the midway positions between R1 and R2 using the following interpolation methods. If doing Euler rotations, assume the rotation is local X then local y then local z (or XYZ in world space). Show a drawing of the stick and card, and also a 3x3 rotation matrix giving its frame orientation.

You can consult together on this assignment but must turn in your own solution.

1. What is the frame orientation if you interpolate using Euler rotations.
2. What is the frame orientation if you interpolate using axis-angle rotation as given in the text.
3. What is the frame orientation if you rotate using quaternions linearly (not "slerping").
4. What is the frame orientation if you rotate using quaternions and "slerp".
5. What rotation applied to the first frame R1 will take you to the second frame R2? Where will the frame be if you don't interpolate between the two frames, but apply half this rotation?
6. Similarly, what is the quaternion that will rotate R1 such that it is aligned with R2? Where will the frame be if you apply half of this quaternion rotation?

Questions?:

E-mail wilhelms@cse.ucsc.edu.