CONVERSION OF COORDINATES BETWEEN FRAMES

Transcription

1 ECS 178 Course Notes CONVERSION OF COORDINATES BETWEEN FRAMES Kenneth I. Joy Institute for Data Analysis and Visualization Department of Computer Science University of California, Davis Overview Frames are used to represent coordinate systems in computer graphics. They define local coordinate systems for objects, for cameras, and form models of various types. A necessary operation in computer graphics is to convert the coordinates of a point in one frame, to coordinates of the point in a second frame. These conversions can be defined via matrices and the actual conversions carried out via matrix multiplication. A Useful Example that Illustrates Conversion of Coordinates Between Frames Consider the two frames ( u 1, v 1, O 1 ) and ( u 2, v 2, O 2 ) in two dimensions, where u 1 =< 1, 0 > u 2 =< 1, 0 > v 1 =< 1, 1 > v 2 =< 0, 2 > O 1 =< 0, 0 > O 2 =< 2, 2 > These frames are shown in the figure below:

2 v 2 O 2 u 2 v 1 O 1 u 1 Suppose we have a point P with coordinates (3, 2) in the frame ( u 1, v 1, O 1 ). What are the coordinates of the point in the frame ( u 2, v 2, O 2 )? If we use the matrix notation for the point P, we have P = 3 u v 1 + O 1 u 1 = v 1 O 1 u 2 = u v 2 2 u 2 v 2 + O u 2 = v O 2 u 2 = v 2 O 2 2

3 and therefore the new coordinate is (3, 0) under the frame ( u 2, v 2, O 2 ). Now carefully note what we did in this calculation. We first wrote the point in the matrix notation for the frame ( u 1, v 1, O 1 ). Then we wrote the vectors of the first frame in terms of the vectors of the second frame (We can do this since the vectors of the second frame form a basis for the space of 2d vectors.). and wrote the origin O 1 in terms of the origin and vectors of the second frame. This column vector, expressing the first frame in terms of the second, was converted to the product of a 3 3 matrix and the column vector representing the second frame. Finally, we multiplied the 3 3 matrix times the initial coordinate to obtain the result. It is easy to see that this can be done with any coordinate from the first frame, as the 3 3 matrix generated will be the same in every case. That is, given a coordinate (u, v) from the first frame, we can convert it to a coordinate for the second frame by simply calculating u v = (u + v 2) ( 1 2 v 1) 1 Conversion Between Arbitrary Frames conversion of coordinates from one frame to another in n-dimensional space proceeds in the same way as the example above. Suppose a point P has coordinates (c 1, c 2,..., c n, 1) relative to some frame F = ( v 1, v 2,..., v n, O). What would be the coordinates of P relative to another frame F = ( v 1, v 2,..., v n, O )? Since the vectors v 1, v 2,..., v n of F form a basis, we can write each of the vectors v 1, v 2,..., v n and O uniquely in terms of v 1, v 2,..., and v n and O. So there are coefficients e i,j defined by v i = e i,1 v 1 + e i,2 v e i,n v n for i = 1, 2,..., n and O = e n+1,1 v 1 + e n+1,2 v e n+1,n v n + O 3

4 and now we can write c 1 v 1 + c 2 v c n v n O = c 1 c 2 c n 1 v 1 v 2. v n O = = c 1 c 2 c n 1 c 1 c 2 c n 1 e 1,1 v 1 + e 1,2 v e 1,n v n e 2,1 v 1 + e 2,2 v e 2,n v n. e n,1 v 1 + e n,2 v e n,n v n e n+1,1 v 1 + e n+1,2 v e n+1,n v n + O e 1,1 e 1,2 e 1,n 0 v 1 e 2,1 e 2,2 e 2,n 0 v e n,1 e n,2 e n,n 0 v n e n+1,1 e n+1,2 e n+1,n 1 O and since the vectors v 1, v 2,..., v n are linearly independent we have that the coordinates (c 1, c 2,..., c n ) of the point in the frame F are given by e 1,1 e 1,2 e 1,n 0 e 2,1 e 2,2 e 2,n 0 c 1 c 2 c n 1 = c 1 c 2 c n e n,1 e n,2 e n,n 0 e n+1,1 e n+1,2 e n+1,n 1 that is, the change of coordinates is accomplished via a (n + 1) (n + 1) matrix multiplication where the rows of the matrix consist of the coordinates of the elements of the old frame F relative to the new frame F. How Do We Calculate the Conversion Matrix 4

5 So how do we calculate the matrix e 1,1 e 1,2 e 1,n 0 e 2,1 e 2,2 e 2,n e n,1 e n,2 e n,n 0 e n+1,1 e n+1,2 e n+1,n 1? We use Cramer s Rule. Since most of the calculations we are required to do are in 3 dimensions, we will present the techniques for calculating the matrix in this special case where the frames will now be denoted as ( u 1, v 1, w 1, O 1 ) and ( u 2, v 2, w 2, O 2 ) and the matrix will be e 1,1 e 1,2 e 1,3 0 e 2,1 e 2,2 e 2,3 0 e 3,1 e 3,2 e 3,3 0 e 4,1 e 4,2 e 4,3 1 How do we find the values e i,j? Given any frame ( u, v, w, O), and a vector t, we know that t can be written as t = u u + v v + w w for some constants u, v and w. Utilizing Cramer s Rule, these constants can be calculated by the following process: If we define D = u ( v w) D 1 = t ( v w) D 2 = u ( t w) D 3 = u ( v t) then u = D 1 D v = D 2 D w = D 3 D 5

6 Utilizing this process, we can let ( u, v, w, O) = ( u 2, v 2, w 2, O 2 ), and first let t = u 1 and calculate e 1,1, e 1,2 and e 1,3 ; then let t = v 1 and calculate e 2,1, e 2,2 and e 2,3 ; let t = w 1 and calculate e 3,1, e 3,2 and finally let t = O 2 O 1 and calculate e 4,1, e 4,2 and e 4,3. Using Cramer s rule, the calculations are straightforward and consist of taking a few cross products and dot products. See the section on Cramer s Rule for more information when the frames are orthonormal. Summary Frames are a useful tool in computer graphics and geometric modeling as they serve as local coordinate systems for objects. The procedures to convert between frames to convert the points defined by one local coordinate system to become points in another local coordinate system are specified by a 4 4 matrix and the conversion is carried out by matrix multiplication. References All contents copyright (c) Computer Science Department, University of California, Davis All rights reserved. 6