Creating a plane/disk perpendicular to a line segment

There are a number of 3D geometric construction techniques that require a coordinate system perpendicular to a line segment, some examples are:

• Creating a disk given its center, radius and normal.
  
• Forming a cylinder given its two end points and radii at each end.
  
• Creating a plane coordinate system perpendicular to a line.
  

A straightforward method will be described which facilitiates each of these. The key is deriving a pair of orthonormal vectors on the plane perpendicular to a line segment P₁, P₂.

Procedure

1. Choose any point P randomly which doesn't lie on the line through P₁ and P₂
2. Calculate the vector R as the cross product between the vectors P - P₁ and P₂ - P₁. This vector R is now perpendicular to P₂ - P₁. (If R is 0 then 1. wasn't satisfied)
3. Calculate the vector S as the cross product between the vectors R and P₂ - P₁. This vector S is now perpendicular to both R and the P₂ - P₁.
4. The unit vectors ||R|| and ||S|| are two orthonormal vectors in the plane perpendicular to P₂ - P₁.

Points on the plane through P₁ and perpendicular to n = P₂ - P₁ can be found from linear combinations of the unit vectors R and S, for example, a point Q might be

Q_x = P_1x + alpha R_x + beta S_x
Q_y = P_1y + alpha R_y + beta S_y
Q_z = P_1z + alpha R_z + beta S_z
for all alpha and beta.

A disk of radius r, centered at P₁, with normal n = P₂ - P₁ is described as follows

Q_x = P_1x + r cos(theta) R_x + r sin(theta) S_x
Q_y = P_1y + r cos(theta) R_y + r sin(theta) S_y
Q_z = P_1z + r cos(theta) R_z + r sin(theta) S_z
for 0 <= theta <= 2 pi

Example

The following is a simple example of a disk and the C source stub that generated it.