Question

I have a sphere in my program and I intend to draw some rectangles over at a distance x from the centre of this sphere. The figure looks something below:

The rectangles are drawn at (x,y,z) points that I have already have in a vector of 3d points.
Let’s say the distance x from centre is 10. Notice the orientation of these rectangles and these are tangential to an imaginary sphere of radius 10 (perpendicular to an imaginary line from the centre of sphere to the centre of rectangle)

Currently, I do something like the following:

For n points vector<vec3f> pointsInSpace where the rectnagles have to be plotted

for(int i=0;i<pointsInSpace.size();++i){
    //draw rectnagle at (x,y,z)
}

which does not have this kind of tangential orientation that I am looking for.
It looked to me of applying roll,pitch,yaw rotations for each of these rectangles and using quaternions somehow to make them tangential as to what I am looking for.
However, it looked a bit complex to me and I wanted to ask about some better method to do this.

Also, the rectangle in future might change to some other shape, so a kind of generic solution would be appreciated.

Answer 1

I think you essentially want the same transformation as would be accomplished with a LookAt() function (you want the rectangle to ‘look at’ the sphere, along a vector from the rectangle’s center, to the sphere’s origin).

If your rectangle is formed of the points:

(-1, -1, 0)
(-1,  1, 0)
( 1, -1, 0)
( 1,  1, 0)

Then the rectangle’s normal will be pointing along Z. This axis needs to be oriented towards the sphere.

So the normalised vector from your point to the center of the sphere is the Z-axis.

Then you need to define a distinct ‘up’ vector – (0,1,0) is typical, but you will need to choose a different one in cases where the Z-axis is pointing in the same direction.

The cross of the ‘up’ and ‘z’ axes gives the x axis, and then the cross of the ‘x’ and ‘z’ axes gives the ‘y’ axis.

These three axes (x,y,z) directly form a rotation matrix.

This resulting transformation matrix will orient the rectangle appropriately. Either use GL’s fixed function pipeline (yuk), in which case you can just use gluLookAt(), or build and use the matrix above in whatever fashion is appropriate in your own code.