Arcball When viewing three dimensional objects there are many different ways to allow a user to choose a viewpoint, but the method I have found to be the most intuitive is the arcball. (For examples of many different methods, visit Chris Rorden's page.) The basic principle of the arcball is this: create a sphere around the object, and allow the user to click a point on the sphere and drag that point to a different location on the screen, having the object rotate to follow it. The following ZIP file contains C++ source code implementing an arcball in OpenGL, the mechanics of which will be explained on this page. It also includes the example Earth program which I used to demonstrate its function, and source code for a small sample program using GLUT. Arcball (C++ and Win32 demo) 5/05/2008, ZIP File 215kB This arcball source code is free to use and modify for any purpose, with no restrictions of copyright or license. If you have any comments or questions, please e-mail them to me. My address appears at the bottom of this webpage.

Finding the Mouse Position

The first step in finding the mouse position is to aquire the viewing transformation matrices from OpenGL. I recommend setting up your view using just the PROJECTION matrix and then acquiring it directly from OpenGL with glGetDoublev. (I'm assuming that the MODELVIEW matrix is identity for the viewer, so I leave it as is.)

GLdouble projection_matrix[16] = {1,0,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1};
GLdouble modelview_matrix[16] = {1,0,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1};
int viewport_matrix[4] = {0,0,640,480};

glGetDoublev(GL_PROJECTION_MATRIX,projection_matrix);
glGetIntegerv(GL_VIEWPORT,viewport_matrix);

Once these matrices have been acquired and stored for future use, we can use them to handle mouse input. Knowing the mouse coordinates mouse_x, and mouse_y, we can use gluUnProject to acquire the coordinates of a point in the scene that is under the mouse. Note, though, that OpenGL counts from the bottom left corner whereas most windows interfaces count from the top left, so you may need to invert your mouse_y coordinates. (e.g. mouse_y = (window_height - 1) - mouse_y;)

GLdouble x,y,z;
gluUnProject(mouse_x, mouse_y, 0.0,
             modelview_matrix,
             projection_matrix,
             viewport_matrix,
             &x, &y, &z);

Once a point in the scene has been found, it is a matter of creating a ray from the camera location to that point, then finding the intersection of that ray with your sphere. Your ray is the set of points E + t*(P - E), where E is the eye location, P is the point in the scene, and t is a variable parameter. Similarly, your sphere is the set of points S where S² = r², r being the radius. If S = E + t*(P - E), then the point exists both on the sphere and the ray, so we can make a substitution to determine that (E + t*(P - E))² = r². I will leave the algebra up to you, but the gist of it is that you know the vectors E, P, and the scalar r, so t can be found by plugging them in. You'll get a quadratic equation which gives from 0 to 2 solutions. If there's no solution, your ray does not intersect the sphere (there are many ways to deal with this, I will not go into them here), and if there are two, you should pick the one that gives the point closest to the eye.

A diagram of the basic principle of the arcball's rotation.

Determining the Arcball's Rotation

If you click and drag one point on the sphere to another location, the question that must be answered is "how do I rotate the sphere to move the original point to the new location?" The perhaps unexpected answer is that there are really an infinite number of ways to do it. Imagine putting your finger on a basketball, then spinning it. Despite the spinning, the point under your finger hasn't moved. If the only qualification you have to meet is to have the original point end up at a new location, how are you to choose between these different possible rotations?

A simple solution can be seen in the diagram to the right. Take the cross product of your start and end points (unit length vectors from the centre of the sphere) to form a rotational axis perpendicular to both of them. Then to find the angle it must be rotated about this axis, take their dot product, which gives you the cosine of that angle.

Calculating the Rotation Matrix

At this point you could just call glRotate knowing the axis of rotation and getting the angle with an arccosine function, but you can also keep track of it yourself. The way to do this is with a special kind of mathematical construction called a quaternion.

A quaternion is an extension of two dimensional complex numbers to four dimensions (there is no corresponding three dimensional type of number, interestingly). Specifically for rotations, though, we can think of our quaternion as a 4D vector (x,y,z,w) where (x,y,z) would point along our rotational axis in 3D, and w is related to our angle. (There is a good description of using quaternions for rotation at Genesis 3D.)

If we have a unit vector (x,y,z), we may obtain the correct magnitude of the quaternion's (x,y,z) by multiplying them by the sine of half the angle of rotation. Finally w is determined to be the cosine of half the angle rotation. Once known, all of these values can be assembled into a rotation matrix that looks like the following:

The generation of this matrix can be slightly simplified knowing that (w² + x² + y² + z²) = 1. If this sum is not equal to 1, the corresponding matrix transformation will actually cause some sort of scaling as well as the rotation, which would generally be quite undesirable. (This is why it is important that your axis of rotation be a unit vector scaled by the sine of the half angle. They combine with its cosine to form a 4D vector that is also of unit length.)

Once you have your rotation matrix, however, you can multiply it by whatever matrix you were using when the starting vector was gathered, and pass the result to glMultMatrix, which will apply the transformation for you.

That's it! There is much more that can be said about quaternions, but that is a big topic and not directly related to the arcball. Take a look at the source code in the ZIP file for clues as to how to implement the thing, but as always if you have questions, send me an e-mail.

brad rainwarrior.ca
5/05/2008