papercraft/tri.c

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/*
* Triangle manipulations
*/
#include <stdio.h>
#include <math.h>
#include <stdint.h>
#include "tri.h"
int tri_debug = 0;
tri_t *
tri_new(
const v3_t * p_cam,
const v3_t * p_xyz
)
{
tri_t * const t = calloc(1, sizeof(*t));
if (!t)
return NULL;
for(int i = 0 ; i < 3 ; i++)
t->p[i] = p_cam[i];
// precompute the normals
t->normal = v3_norm(v3_cross(
v3_sub(t->p[1], t->p[0]),
v3_sub(t->p[2], t->p[1])
));
t->normal_xyz = v3_norm(v3_cross(
v3_sub(p_xyz[1], p_xyz[0]),
v3_sub(p_xyz[2], p_xyz[1])
));
// compute the bounding box for the triangle in camera space
t->min = v3_min(v3_min(t->p[0], t->p[1]), t->p[2]);
t->max = v3_max(v3_max(t->p[0], t->p[1]), t->p[2]);
return t;
}
// insert a triangle into our z-sorted list
void
tri_insert(
tri_t ** zlist,
tri_t * t
)
{
while(1)
{
tri_t * const iter = *zlist;
if (!iter)
break;
// check to see if our new triangle is closer than
// the current triangle
if(iter->min.p[2] > t->min.p[2])
break;
zlist = &(iter->next);
}
// either we reached the end of the list,
// or we have found where our new triangle is sorted
t->next = *zlist;
t->prev = zlist;
*zlist = t;
if (t->next)
t->next->prev = &t->next;
}
void
tri_delete(tri_t * t)
{
if (t->next)
t->next->prev = t->prev;
if (t->prev)
*(t->prev) = t->next;
t->next = NULL;
t->prev = NULL;
free(t);
}
// Compute the 2D area of a triangle in screen space
// using Heron's formula
float
tri_area_2d(
const tri_t * const t
)
{
const float a = v3_dist_2d(&t->p[0], &t->p[1]);
const float b = v3_dist_2d(&t->p[1], &t->p[2]);
const float c = v3_dist_2d(&t->p[2], &t->p[0]);
const float s = (a + b + c) / 2;
return sqrt(s * (s-a) * (s-b) * (s-c));
}
void
tri_print(
const tri_t * const t
)
{
fprintf(stderr, "{{{%+5.1f,%+5.1f,%5.1f }},{{%+5.1f,%+5.1f,%5.1f }},{{%+5.1f,%+5.1f,%5.1f }}}\n", // norm %.3f %.3f %.3f\n",
t->p[0].p[0],
t->p[0].p[1],
t->p[0].p[2],
t->p[1].p[0],
t->p[1].p[1],
t->p[1].p[2],
t->p[2].p[0],
t->p[2].p[1],
t->p[2].p[2]
//t->normal.p[0],
//t->normal.p[1],
//t->normal.p[2]
);
}
/* Check if two triangles are coplanar and share an edge.
*
* Returns -1 if not coplanar, 0-2 for the edge in t0 that they share.
*/
int
tri_coplanar(
const tri_t * const t0,
const tri_t * const t1,
const float coplanar_eps
)
{
// the two normals must be parallel-enough
const float angle = v3_mag(v3_sub(t0->normal_xyz, t1->normal_xyz));
if (angle < -coplanar_eps || +coplanar_eps < angle)
return -1;
// find if there are two points shared
unsigned matches = 0;
for(int i = 0 ; i < 3 ; i++)
{
for(int j = 0 ; j < 3 ; j++)
{
if (!v3_eq(t0->p[i], t1->p[j]))
continue;
matches |= 1 << i;
break;
}
}
switch(matches)
{
case 0x3: return 0;
case 0x6: return 1;
case 0x5: return 2;
case 0x7:
// these are likely small triangles that can be ignored
if (tri_debug > 3)
{
fprintf(stderr, "uh, three points match?\n");
tri_print(t0);
tri_print(t1);
}
return 0;
default:
// no shared edge
return -1;
}
}
/*
* Find the Z point of an XY coordinate in a triangle.
*
* p can be written as a combination of t01 and t02,
* p - t0 = a * (t1 - t0) + b * (t2 - t0)
* setting t0 to 0, this becomes:
* p = a * t1 + b * t2
* which is two equations with two unknowns
*
* Returns true if the point is inside the triangle
*/
int
tri_find_z(
const tri_t * const t,
const v3_t * const p,
float * const zout
)
{
const float t1x = t->p[1].p[0] - t->p[0].p[0];
const float t1y = t->p[1].p[1] - t->p[0].p[1];
const float t1z = t->p[1].p[2] - t->p[0].p[2];
const float t2x = t->p[2].p[0] - t->p[0].p[0];
const float t2y = t->p[2].p[1] - t->p[0].p[1];
const float t2z = t->p[2].p[2] - t->p[0].p[2];
const float px = p->p[0] - t->p[0].p[0];
const float py = p->p[1] - t->p[0].p[1];
const float a = (px * t2y - py * t2x) / (t1x * t2y - t2x * t1y);
const float b = (px * t1y - py * t1x) / (t2x * t1y - t1x * t2y);
const float z = t->p[0].p[2] + a * t1z + b * t2z;
if (zout)
*zout = z;
return 0 <= a && 0 <= b && a + b <= 1;
}
/*
* Find the barycentric coordinates for point of an XY coordinate in a triangle.
*
* p can be written as a combination of t01 and t02,
* p - t0 = a * (t1 - t0) + b * (t2 - t0)
* setting t0 to 0, this becomes:
* p = a * t1 + b * t2
* which is two equations with two unknowns
*
* The x and y coordinates are based on the two sides of the triangle
* and the z coordinate is the screen coordinate z in the triangle.
*/
v3_t
tri_bary_coord(
const tri_t * const t,
const v3_t * const p
)
{
const float t1x = t->p[1].p[0] - t->p[0].p[0];
const float t1y = t->p[1].p[1] - t->p[0].p[1];
const float t1z = t->p[1].p[2] - t->p[0].p[2];
const float t2x = t->p[2].p[0] - t->p[0].p[0];
const float t2y = t->p[2].p[1] - t->p[0].p[1];
const float t2z = t->p[2].p[2] - t->p[0].p[2];
const float px = p->p[0] - t->p[0].p[0];
const float py = p->p[1] - t->p[0].p[1];
float a = (px * t2y - py * t2x) / (t1x * t2y - t2x * t1y);
float b = (px * t1y - py * t1x) / (t2x * t1y - t1x * t2y);
v3_t v = {{
a,
b,
t->p[0].p[2] + a * t1z + b * t2z,
}};
//v.p[2] = 1.0 - v.p[0] - v.p[1];
return v;
}
// Returns true if the bary centry point is inside the triangle
// which means that a and b are non-negative and sum to less than 1
int
tri_bary_inside(
const v3_t p
)
{
const float a = p.p[0];
const float b = p.p[1];
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return 0 <= a && 0 <= b && a + b <= 1+EPS;
}
/** Compute the points of intersection for two segments in 2d, and z points.
*
* This is a specialized ray intersection algorithm for the
* hidden wire-frame removal code that computes the intersection
* points for two rays (in 2D, "orthographic") and then computes
* the Z depth for the intersections along each of the segments.
*
* Returns -1 for non-intersecting, otherwise a ratio of how far
* along the intersection is on the l0.
*/
float
hidden_intersect(
const v3_t * const p0,
const v3_t * const p1,
const v3_t * const p2,
const v3_t * const p3,
v3_t * const l0_int,
v3_t * const l1_int
)
{
const float p0_x = p0->p[0];
const float p0_y = p0->p[1];
const float p0_z = p0->p[2];
const float p1_x = p1->p[0];
const float p1_y = p1->p[1];
const float p1_z = p1->p[2];
const float p2_x = p2->p[0];
const float p2_y = p2->p[1];
const float p2_z = p2->p[2];
const float p3_x = p3->p[0];
const float p3_y = p3->p[1];
const float p3_z = p3->p[2];
const float s1_x = p1_x - p0_x;
const float s1_y = p1_y - p0_y;
const float s2_x = p3_x - p2_x;
const float s2_y = p3_y - p2_y;
// compute r x s
const float d = -s2_x * s1_y + s1_x * s2_y;
// if they are close to parallel, then we do not need to check
// for intersection (we define that as "non-intersecting")
if (-EPS < d && d < EPS)
return -1;
// Compute how far along each line they would interesect
const float r0 = ( s2_x * (p0_y - p2_y) - s2_y * (p0_x - p2_x)) / d;
const float r1 = (-s1_y * (p0_x - p2_x) + s1_x * (p0_y - p2_y)) / d;
// if they are not within the ratio (0,1), then the intersecton occurs
// outside of the segments and is not of concern
if (r0 < 0 || r0 > 1)
return -1;
if (r1 < 0 || r1 > 1)
return -1;
// Collision detected with the segments
if(0) fprintf(stderr, "collision: %.0f,%.0f,%.0f->%.0f,%.0f,%.0f %.0f,%.0f,%.0f->%.0f,%.0f,%.0f == %.3f,%.3f\n",
p0_x, p0_y, p0_z,
p1_x, p1_y, p1_z,
p2_x, p2_y, p2_z,
p3_x, p3_y, p2_z,
r0,
r1
);
// compute the z intercept for each on the two different coordinates
if(l0_int)
{
*l0_int = (v3_t){{
p0_x + r0 * s1_x,
p0_y + r0 * s1_y,
p0_z + r0 * (p1_z - p0_z)
}};
}
if(l1_int)
{
*l1_int = (v3_t){{
p2_x + r1 * s2_x,
p2_y + r1 * s2_y,
p2_z + r1 * (p3_z - p2_z)
}};
}
return r0;
}
/*
* Fast check to see if t2 is entire occluded by t.
*/
int
tri_behind(
const tri_t * const t,
const tri_t * const t2
)
{
float z0, z1, z2;
int inside0 = tri_find_z(t, &t2->p[0], &z0);
int inside1 = tri_find_z(t, &t2->p[1], &z1);
int inside2 = tri_find_z(t, &t2->p[2], &z2);
// easy check -- if none of the points are inside,
// t2 is not entirely occluded
if (!inside0 || !inside1 || !inside2)
return 0;
// are all of the intersection points ahead of t2?
int behind0 = t2->p[0].p[2] >= z0;
int behind1 = t2->p[1].p[2] >= z1;
int behind2 = t2->p[2].p[2] >= z2;
if (behind0 && behind1 && behind2)
return 1;
// it is a STL violation if they are not all on the
// same side (this would indicate that t and t2 intersect
// go ahead and prune since it will cause problems
if (behind0 || behind1 || behind2)
{
/*
fprintf(stderr, "WARNING: triangles intersect %.0f %.0f %.0f inside %d %d %d behind %d %d %d\n", z0, z1, z2, inside0, inside1, inside2, behind0, behind1, behind2);
tri_print(t);
tri_print(t2);
*/
return 1;
}
// they are all on the same side
return 0;
}
/*
tri_no_intersection, // nothing changed
tri_infront, // segment is in front of the triangle
tri_hidden, // segment is completely occluded
tri_clipped, // segment is partially occluded on one end
tri_split, // segment is partially occluded in the middle
*/
tri_intersect_t
tri_seg_intersect(
const tri_t * t,
seg_t * s,
seg_t ** new_seg // only if tri_split
)
{
// avoid processing nearly empty segments
const float seg_len = v3_len(&s->p[0], &s->p[1]);
if (seg_len < EPS)
return tri_hidden;
const v3_t p_max = v3_max(s->p[0], s->p[1]);
const v3_t p_min = v3_min(s->p[0], s->p[1]);
// if the segment is closer than the triangle,
// then we no longer have to check any further into
// the zlist (it is sorted by depth).
if (p_max.p[2] <= t->min.p[2])
return tri_infront;
// check for four quadrant outside the bounding box
// of the triangle min/max, which would have no chance
// of intersecting with the triangle
if (p_min.p[0] < t->min.p[0]
&& p_max.p[0] < t->min.p[0])
return tri_no_intersection;
if (p_min.p[1] < t->min.p[1]
&& p_max.p[1] < t->min.p[1])
return tri_no_intersection;
if (p_min.p[0] > t->max.p[0]
&& p_max.p[0] > t->max.p[0])
return tri_no_intersection;
if (p_min.p[1] > t->max.p[1]
&& p_max.p[1] > t->max.p[1])
return tri_no_intersection;
// there is a possibility that this line crosses the triangle
// compute the coordinates in triangle space
const v3_t tp0 = tri_bary_coord(t, &s->p[0]);
const v3_t tp1 = tri_bary_coord(t, &s->p[1]);
// if both are inside and not both on the same edge of
// the triangle, then the segment is totally hidden.
if (tri_bary_inside(tp0) && tri_bary_inside(tp1))
{
// if the segment z is closer than the triangle z
// then the segment is in front of the triangle
if (s->p[0].p[2] < tp0.p[2] && s->p[1].p[2] < tp1.p[2])
return tri_no_intersection;
// if the barycentric coord is 0 for the same edge
// for both points, then it is on the original line
if (tp0.p[0] < EPS && tp1.p[0] < EPS)
return tri_no_intersection;
if (tp0.p[1] < EPS && tp1.p[1] < EPS)
return tri_no_intersection;
// compute the third barycentric coordinate and check
float c0 = 1.0 - tp0.p[0] - tp0.p[1];
float c1 = 1.0 - tp1.p[0] - tp1.p[1];
if (c0 < EPS && c1 < EPS)
return tri_no_intersection;
// it is not on an edge and not infront of the triangle
// so the segment is totally occluded
return tri_hidden;
}
// find the intersection point for each of the three
// sides of the triangle
v3_t is[3] = {}; // 3d point of segment intercept
v3_t it[3] = {}; // 3d point of triangle intercept
float ratio[3] = {}; // length along the line
int intersections = 0;
for(int j = 0 ; j < 3 ; j++)
{
ratio[intersections] = hidden_intersect(
&s->p[0], &s->p[1],
&t->p[j], &t->p[(j+1)%3],
&is[intersections],
&it[intersections]
);
if (ratio[intersections] < 0)
continue;
// if the segment intersection is closer than the
// triangle intersection, this does not count as
// an intersection and we can ignore it.
if (is[intersections].p[2] < it[intersections].p[2])
continue;
if (tri_debug >= 2)
{
fprintf(stderr, "%d ratio=%.2f %+6.1f", j, ratio[intersections], it[intersections].p[2]);
v3_print(is[intersections]);
}
intersections++;
}
// check for duplicate intersections, which happens if
// the lines go through at precisely the corners
// this might mean that we hit exactly at one
// point and two of the points are the same
if (intersections == 3)
{
if (v3_eq(is[0], is[2]))
intersections--;
else
if (v3_eq(is[1], is[2]))
intersections--;
else
if (v3_eq(is[0], is[1]))
{
intersections--;
is[1] = is[2];
it[1] = it[2];
}
}
if (intersections == 2 && v3_eq(is[0], is[1]))
intersections--;
// no intersections? there is nothing to do
if (intersections == 0)
return tri_no_intersection;
// three intersections? maybe a very small triangle
if (intersections == 3)
{
fprintf(stderr, "THREE INTERSECTIONS?\n");
fprintf(stderr, "is0="); v3_print(is[0]);
fprintf(stderr, "is1="); v3_print(is[1]);
fprintf(stderr, "is2="); v3_print(is[2]);
svg_line("#0000FF", s->p[0].p, s->p[1].p, 8);
return tri_no_intersection;
}
if (intersections == 1)
{
if (tri_bary_inside(tp0))
{
// if the intercept point on the segment is
// closer than the intercept point on the triangle edge,
// then there is no occlusion
if (is[0].p[2] <= it[0].p[2])
return tri_no_intersection;
// clipped from intersection to p1
s->p[0] = is[0];
return tri_clipped;
}
if (tri_bary_inside(tp1))
{
// if the intercept point on the segment is
// closer than the intercept point on the triangle edge,
// then there is no occlusion
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if (is[0].p[2] <= it[0].p[2])
return tri_no_intersection;
// clipped from p0 to intersection
s->p[1] = is[0];
return tri_clipped;
}
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// maybe we have a small triangle or a tangent.
// in which case the line is fine
return tri_no_intersection;
/*
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fprintf(stderr, "ONE INTERSECTION?");
svg_line("#FFFF00", s->p[0].p, s->p[1].p, 20);
svg_line("#000080", t->p[0].p, t->p[1].p, 8);
svg_line("#000080", t->p[1].p, t->p[2].p, 8);
svg_line("#000080", t->p[2].p, t->p[0].p, 8);
seg_print(s);
v3_print(tp0);
v3_print(tp1);
tri_print(t);
*new_seg = seg_new(is[0], s->p[1]);
s->p[1] = is[0];
return tri_split;
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*/
}
// two intersections: find the one that is closer to p0
// modify the existing segment and create a new segment
const float d00 = v3_mag2(v3_sub(is[0], s->p[0]));
const float d01 = v3_mag2(v3_sub(is[1], s->p[0]));
const float d10 = v3_mag2(v3_sub(is[0], s->p[1]));
const float d11 = v3_mag2(v3_sub(is[1], s->p[1]));
// if any of the intersections points are zero from an
// end point on the segment, then skip that part
if (tri_debug > 4)
{
seg_print(s);
tri_print(t);
fprintf(stderr, "d: %f %f %f %f\n", d00, d01, d10, d11);
}
if (d00 < EPS && d11 < EPS)
return tri_hidden;
if (d01 < EPS && d10 < EPS)
return tri_hidden;
if (d00 < EPS)
{
s->p[0] = is[1];
return tri_clipped;
} else
if (d01 < EPS)
{
s->p[0] = is[0];
return tri_clipped;
} else
if (d10 < EPS)
{
s->p[1] = is[1];
return tri_clipped;
} else
if (d11 < EPS)
{
s->p[1] = is[0];
return tri_clipped;
}
// neither end points match, so create a new segment
// that excludes the space covered by the triangle.
// determine which is closer to point is[0]
if (d00 < d01)
{
// p0 is closer to is0, so new segment is is1 to p1
*new_seg = seg_new(is[1], s->p[1]);
s->p[1] = is[0];
} else {
// p0 is closer to is1, so new segment is is0 to p1
*new_seg = seg_new(is[0], s->p[1]);
s->p[1] = is[1];
}
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if (tri_debug > 3)
{
fprintf(stderr, "SPLIT: ");
seg_print(*new_seg);
}
return tri_split;
}
int
tri_seg_hidden(
const tri_t * zlist,
seg_t * s,
seg_t ** slist_visible
)
{
int count = 0;
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if (tri_debug > 2)
{
fprintf(stderr, "TEST: ");
seg_print(s);
}
for( const tri_t * t = zlist ; t ; t = t->next )
{
seg_t * new_seg = NULL;
tri_intersect_t type = tri_seg_intersect(t, s, &new_seg);
// if there is no intersection or if the segment has
// been clipped on one side, keep looking
if (type == tri_no_intersection)
continue;
if (type == tri_clipped)
{
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//seg_print(s);
continue;
}
// if this segment is infront of this triangle then we can
// stop searching
if (type == tri_infront)
break;
// if this segment is totally occluded, we're done
if (type == tri_hidden)
return count;
// if this line has been split into two, process the
// new segment starting at the next triangle since it
// has already intersected this one
if (type == tri_split)
{
static int recursive;
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int new_count = tri_seg_hidden(
t->next,
new_seg,
slist_visible
);
count += new_count;
continue;
}
fprintf(stderr, "unknown type %d\n", type);
return -1;
}
// we've reached the end and it is still visible
s->next = *slist_visible;
*slist_visible = s;
return ++count;
}