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Linear Algebra and the C Language/a05y
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The coefficients of the equation of a line.
Presentation :
A homogeneous linear system with as many equations
as unknowns has a nontrivial solution if and only
if the determinant of the matrix is zero.
Let us calculate the equation of the line passing through points P and Q:
c1 x + c2 y + c3 = 0
This same equation with the points P(x1,y1) Q(x2,y2):
c1 x1 + c2 y1 + c3 = 0
c1 x2 + c2 y2 + c3 = 0
The system of three equations:
c1 x + c2 y + c3 = 0
c1 x1 + c2 y1 + c3 = 0
c1 x2 + c2 y2 + c3 = 0
The determinant of the system:
|x y 1|
|x1 y1 1| = 0
|x2 y2 1|
The determinant in C language:
|1 1 1|
|x1 y1 1| = 0
|x2 y2 1|
To calculate the coefficients of the equation of the line, we use the cofactor expansion along the first row.
cofactor_R(A,R1,C1) x + cofactor_R(A,R1,C2) y + cofactor_R(A,R1,C3) = 0
This equation gives us the equation of the line
that passes through the two points P and Q.
The application
dline.h .............. Temporary h file
Delete this h file at the end of the session.
c00a.c
c00b.c