Raytrace of a Star SapphireDate: 12/11/96 at 21:04:33 From: Charles Subject: Angle from one 3-dimensional vector to another in terms of rotating on x,y,and z axis. Hello, I am a HS graduate with no college math who is _very_ rusty on pre- calc and geometry/trig. My problem comes from an attempt to create a raytrace of a star sapphire that acts like a real star sapphire, which always faces the camera. To do this, I need to find the degrees of rotation to use to face the star towards the camera position no matter where it is. If memory serves, (which is questionable) the following formula _should_ apply: rotation on x axis = degrees (atan((CamY-ObjY)/(CamZ-ObjZ))) rotation on y axis = degrees (atan((CamZ-ObjZ)/(CamX-ObjX))) rotation on z axis = degrees (atan((CamX-ObjX)/(CamY-ObjY))) Where: CamX,Y,Z = 3-dimensional vector of Camera ObjX,Y,Z = 3-dimensional vector of Object Rotation refers to star on the sapphire. I just can't seem to get the results I want from this. I remember something about the tangent of 90 causing problems, but I'm not too sure. I will really appreciate it if you can help clear this up for me. Thanks... Date: 12/12/96 at 09:39:01 From: Doctor Jerry Subject: Re: Angle from one 3-dimensional vector to another in terms of rotating on x,y,and z axis. Hi Charles, I may be able to help you, but I've got to get the problem clear in terms I can understand and visualize. From your comments above, here are my impressions and, such as they are, my understanding: You have a camera at, say, (a,b,c), which can be thought of as a point or a position vector from the origin. There is an object at the point (A,B,C). There is a plane (the plane of the star) going through the point (A,B,C). My understanding starts to fade as I think about the plane. A plane containing (A,B,C) can have any orientation. The orientation of a plane is usually specified by its normal vector. This is a vector that is perpendicular (normal) to the plane. You appear to be saying that the orientation of the plane, the normal vector to the plane, is the vector {A,B,C}. In any case, if {n1,n2,n3} is a normal to a plane through (A,B,C), do you want to know how to aim the camera so that the film plane is parallel to the plane with normal {n1,n2,n3}? Specifically, do you want the angles the normal vector makes with the x-, y-, and z-axes? -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 12/12/96 at 12:51:42 From: Anonymous Subject: Re: Angle from one 3-dimensional vector to another in terms of rotating on x,y,and z axis. Sorry for being unclear. The object is originally created at (0,0,0) with a surface normal for the star being positive y. The object is then translated to any other position, (1,0,3) for example, after which, I need the star texture on it to rotate in such a way that the surface normal now faces directly at the camera, at location (2,0,-1). I am trying to find out what formula to use to determine the rotations necessary (in degrees) first on the x, then the y, and finally the z axis in order to face the _texture_ towards the camera at all times, no matter the position of the object or the camera. I hope this clears things up a bit. I am very grateful for your assistance. Charles Date: 12/14/96 at 13:24:54 From: Doctor Jerry Subject: Re: Angle from one 3-dimensional vector to another in terms of rotating on x,y,and z axis. Hi Charles, I solved a sample problem (see below), which may help you decide if I'm on the right track and, if I am, whether you want to pursue this approach. Before we get to this, I need to make a comment or two. First, although I haven't solved this kind of problem before and don't find a full solution in the books I own, I'm certain that the problem has been solved many times before. There are individuals who could help you solve your problem more rapidly and efficiently than I. To act on this, I'll post a memo on the Math Forum. Someone may offer to help. I think that your problem boils down to this: If u and v are unit vectors based at the origin, you want to: (i) rotate v about the x-axis, giving a new vector p (ii) rotate p about the y-axis, giving a vector q (iii) rotate q about the z-axis, giving a vector r, such that r = u. I took u = {1/Sqrt[6],1/Sqrt[6],-2/Sqrt[6]} v = {1/Sqrt[21],4/Sqrt[21],2/Sqrt[21]} Since u and v are unit vectors, their third components give the cosines of the angles these vectors make with the z-axis. I visualized the vector u as lying on a cone whose vertex is at the origin and vertical axis. The third component of u determines the cone. I first rotate v about the x-axis until it lies on the upper nappe (in this case) of the cone. This gives p. Next, I rotate p about the y-axis until it lies on the lower nappe of the cone. This gives q. Finally, I rotate q about the z-axis until it coincides with u. Each rotation is done using a rotation matrix. These matrices are: M1(a) = 1 0 0 0 cos a -sin a 0 sin a cos a M2(b) = cos b 0 -sin b 0 1 0 sin b 0 cos b M3(c) = cos c -sin c 0 sin c cos c 0 0 0 1 I choose a so that the third component of M1(a)*v is equal to the absolute value of the third component of u. I got a = 0.527509 in radians. Then p = {0.218218,0.534523,0.816496}. I choose b so that the third component of M2(b)*p is equal to the third component of u. I got b = pi. Then q = {-0.218217,0.534523,-0.816497}. Finally, I choose c so that the first component of M3(c)*q is equal to the first component of u. I got c = 3.53939. Then r={0.408248,-0.408249,-0.816497}, which is u. There were several decision points in this procedure, requiring some visualization. I graphed the vectors as I went. I used Mathematica for my calculations and graphs. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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