Casio fx-CG50: Intersection of Two Planes
Introduction
The program INTPLACE calculates a possible line:
r = V + t * D
where V = constant vector, D = directional vector, r = linear vector
The planes are have the equations:
A * X + B * Y + C * Z = D
E * X + F * Y + G * Z = H
The directional vector is calculated by the cross product of vectors:
[ A, B, C ] × [ E, F, G ]
= < B * F – E * C, -(A * F – C * D), A * E – B * D >
A system of equations is solved for two of the variables, with the third variable, often z, designated an arbitrary value, often 0.
Casio fx-CG 50 Program: INTPLACE
Bytes: 312
Due to the mixed use of vectors and matrices on the calculator, I have the program take inputs and store them in variables A through H, so we have the greatest flexibility.
The planes have the equations:
A * X + B * Y + C * Z = D
E * X + F * Y + G * Z = H
There is a for loop to simulate a small wait cycle. There is no Wait command in Casio basic.
Code:
For 1 → I To 25
Locate 1, 2, “INTERSECT 2 PLANES”
Blue Locate 1, 4, “AX+BY+CZ=D”
Red Locate 1, 5, “EX+FY+GZ=H”
Next
ClrText
“A”? → A
“B”? → B
“C”? → C
“D”? → D
“E”? → E
“F”? → F
“G”? → G
“H”? → H
CrossP([[A,B,C],[[E,F,G]]) → Vct D
[[A,B][E,F]]^-1 × [[D][H]] → VctH
Mat V [1,1] → I
Mat V [2, 1] → J
[[I, J, 0]] → Mat V
ClrText
“Vct V + T × Vct D”
“0 ≤ T ≤ 1” ◢
“Vct V” ◢
Vct V ◢
“Vct D” ◢
Vct D
The program assumes that z-component of the calculated directional vector (Vct D) is nonzero. The z-component of the constant vector (Vct V) is assigned to be 0.
Example
3 * X – 5 * Y + Z = 4
X – 2 * Y + Z = 0
A = 3
B = -5
C = 1
D = 4
E = 1
F = -2
G = 1
H = 0
Constant Vector:
Vct V = < 8, 4, 0 >
Directional Vector:
Vct D = < -3, -2, -1 >
A line that intersects plane is:
r(t) = < 8, 4, 0 > + t * < -3, -2, -1 >
for 0 ≤ t ≤ 1
Source
Tremblay, Christopher. Mathematics for Game Developers. Thomson Course Technology. Boston, MA. 2004. ISBN 1-59200-038-X. pp. 103 – 105.
Eddie
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