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# Tetrahedron
# Tetrahedron bounding box size:
# Figure 1: Side View
#
# *_ -
# / \ ^
# / | \_
# / | \ depth
# / | \_ v
# *----0-------* -
#
# Figure 2: Top View
#
# *_B
# | \_
# |\ \_
# | \ \_
# | \ \_
# | \A \__
# |<a> *-<b>-__*D
# | / _/
# | / _/
# | / _/
# |/ _/
# |_/
# *C
# From these diagrams, you can tell that the width is the height of an
# equilateral triangle. If we assume the edge of the tetrahedron is s, then:
define @ns s - 4
define @half_height s / 2
define @nhalf_height @ns / 2
# For the depth, a few calculations are necessary:
#
# a + b = s * sin(⅓π)
# a² + (s/2)² = b²
define @a (s/2)*tan(⅙π)
define @na (@ns/2)*tan(⅙π)
define @b s * (sin(⅓π) - tan(⅙π)/2)
define @nb @ns * (sin(⅓π) - tan(⅙π)/2)
# a² + d² = (s*sin(⅓π))²
define @depth s * sqrt((sin(⅓π))^2 - ((1/2)*tan(⅙π))^2)
define @ndepth @ns * sqrt((sin(⅓π))^2 - ((1/2)*tan(⅙π))^2)
# Considering that the origin is in the center of the base, the min x is -a,
# the min y is -s/2, and the min z is 0
-@a < x < @b
-@half_height < y < @half_height
0 < z < @depth
# A can be given by:
#
# (0,0,depth)
# B and C can be given by:
#
# (-a,±half_height,0)
# D can be given by:
#
# (b,0,0)
# The tetrahedron can be given by the intersection of 4 semi-spaces.
# However, one of the semi-spaces is z >= 0, which is implicit in the
# declaration of the z boundaries, so it will not require a condition.
# Cross Product Cheatsheet
#
# a ⨯ b = | i j k |
# | ax ay az |
# | bx by bz |
#
# = (ay bz - az by, az bx - ax bz, ax by - ay bx)
# The normal vector for the plane ABC is the cross product of AB and AC
#
# AB = (-a, half_height, -depth)
# AC = (-a, -half_height, -depth)
#
# vABC = (half_height * (-depth) - (-depth) * (-half_height),
# (-depth) * (-a) - (-a) * (-depth),
# (-a) * (-half_height) - half_height * (-a)) =
# (- s * depth, 0, s * a)
# The normal vector for the plane ADB is the cross product of AD and AB
#
# AD = (b,0,-depth)
# AB = (-a, half_height, -depth)
#
# vABD = (depth * half_height, depth * a + b * depth, b * half_height)
# The normal vector for the plane ACD is equal to the previous one but
# reflected on the y axis. Thus, we only have to flip a plus to a minus.
{x * (- s * @depth) + (z - @depth) * (s * @a) ≤ 0 ∧ \
x * @depth * @half_height + y * @depth * (@a + @b) + (z - @depth) * @b * @half_height ≤ 0 ∧ \
x * @depth * @half_height - y * @depth * (@a + @b) + (z - @depth) * @b * @half_height ≤ 0} \
⊻ \
{x * (- @ns * @ndepth) + (z - @ndepth) * (@ns * @na) ≤ 0 ∧ \
x * @ndepth * @nhalf_height + y * @ndepth * (@na + @nb) + (z - @ndepth) * @nb * @nhalf_height ≤ 0 ∧ \
x * @ndepth * @nhalf_height - y * @ndepth * (@na + @nb) + (z - @ndepth) * @nb * @nhalf_height ≤ 0 ∧ \
z > 1}