Linear dependence means that one linear equation, vector, or related object can be expressed using a linear combination of others. In a system, dependent equations do not add an independent constraint: they may describe the same hyperplane, permit infinitely many solutions, or reveal that the geometry has collapsed. section 3.2.1 section 6.3.1
Recognize linear dependence through its practical consequences:
- In an overspecified system, coincident or dependent hyperplanes can allow a solution even when there are more equations than variables; numerical methods may also use redundant measurements to reduce error. section 3.2.1
- For polynomial interpolation, a point lying on the line determined by two others creates a dependent equation, allowing a lower-degree polynomial to fit more points than usual. section 6.3.1
- In ray-triangle intersection, a zero divisor signals dependence caused by a ray parallel to the triangle’s plane, a degenerate triangle, or a zero-length direction vector. section 3.8.3
Ray-triangle intersection
Write the ray and triangle equations as a three-variable linear system for t, u, and v. The common denominator div measures whether the system is independent enough to solve. Check div before computing the parameters; if div is zero, return no intersection. Otherwise, accept the result only when t ≥ 0, u ≥ 0, v ≥ 0, and u + v ≤ 1. section 3.8.3
Listing 3.1 Solving the ray-triangle intersection symbolically
from sympy import *
Px, Py, Pz = symbols('Px Py Pz') #1
dx, dy, dz = symbols('dx dy dz') #2
Ax, Ay, Az = symbols('Ax Ay Az') #3
ABx, ABy, ABz = symbols('ABx ABy ABz') #4
ACx, ACy, ACz = symbols('ACx ACy ACz') #5
t, u, v = symbols('t u v') #6
solution = solve([
Px +t*dx - (Ax + ABx*u + ACx*v),
Py +t*dy - (Ay + ABy*u + ACy*v),
Pz +t*dz - (Az + ABz*u + ACz*v)
], (t, u, v))
print (pycode(solution))
A SymPy listing constructs and solves the ray-triangle linear system symbolically, exposing the shared divisor used to detect dependence before parameter computation. section 3.8.3
