ref: d35e4b88e85d9bb4f677083c0808b02f51620c2f
dir: /penrose.c/
/*
* Generate Penrose tilings via combinatorial coordinates.
*
* For general explanation of the algorithm:
* https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-tilings/
*
* I use exactly the same indexing system here that's described in the
* article. For the P2 tiling, acute isosceles triangles (half-kites)
* are assigned letters A,B, and obtuse ones (half-darts) U,V; for P3,
* acute triangles (half of a thin rhomb) are C,D and obtuse ones
* (half a thick rhomb) are X,Y. Edges of all triangles are indexed
* anticlockwise around the triangle, with 0 being the base and 1,2
* being the two equal legs.
*/
#include <assert.h>
#include <stddef.h>
#include <string.h>
#include "puzzles.h"
#include "penrose.h"
#include "penrose-internal.h"
#include "tree234.h"
bool penrose_valid_letter(char c, int which)
{
switch (c) {
case 'A': case 'B': case 'U': case 'V':
return which == PENROSE_P2;
case 'C': case 'D': case 'X': case 'Y':
return which == PENROSE_P3;
default:
return false;
}
}
/*
* Result of attempting a transition within the coordinate system.
* INTERNAL means we've moved to a different child of the same parent,
* so the 'internal' substructure gives the type of the new triangle
* and which edge of it we came in through; EXTERNAL means we've moved
* out of the parent entirely, and the 'external' substructure tells
* us which edge of the parent triangle we left by, and if it's
* divided in two, which end of that edge (-1 for the left end or +1
* for the right end). If the parent edge is undivided, end == 0.
*
* The type FAIL _shouldn't_ ever come up! It occurs if you try to
* compute an incoming transition with an illegal value of 'end' (i.e.
* having the wrong idea of whether the edge is divided), or if you
* refer to a child triangle type that doesn't exist in that parent.
* If it ever happens in the production code then an assertion will
* fail. But it might be useful to other users of the same code.
*/
typedef struct TransitionResult {
enum { INTERNAL, EXTERNAL, FAIL } type;
union {
struct {
char new_child;
unsigned char new_edge;
} internal;
struct {
unsigned char parent_edge;
signed char end;
} external;
} u;
} TransitionResult;
/* Construction function to make an INTERNAL-type TransitionResult */
static inline TransitionResult internal(char new_child, unsigned new_edge)
{
TransitionResult tr;
tr.type = INTERNAL;
tr.u.internal.new_child = new_child;
tr.u.internal.new_edge = new_edge;
return tr;
}
/* Construction function to make an EXTERNAL-type TransitionResult */
static inline TransitionResult external(unsigned parent_edge, int end)
{
TransitionResult tr;
tr.type = EXTERNAL;
tr.u.external.parent_edge = parent_edge;
tr.u.external.end = end;
return tr;
}
/* Construction function to make a FAIL-type TransitionResult */
static inline TransitionResult fail(void)
{
TransitionResult tr;
tr.type = FAIL;
return tr;
}
/*
* Compute a transition out of a triangle. Can return either INTERNAL
* or EXTERNAL types (or FAIL if it gets invalid data).
*/
static TransitionResult transition(char parent, char child, unsigned edge)
{
switch (parent) {
case 'A':
switch (child) {
case 'A':
switch (edge) {
case 0: return external(2, -1);
case 1: return external(0, 0);
case 2: return internal('B', 1);
}
case 'B':
switch (edge) {
case 0: return internal('U', 1);
case 1: return internal('A', 2);
case 2: return external(1, +1);
}
case 'U':
switch (edge) {
case 0: return external(2, +1);
case 1: return internal('B', 0);
case 2: return external(1, -1);
}
default:
return fail();
}
case 'B':
switch (child) {
case 'A':
switch (edge) {
case 0: return internal('V', 2);
case 1: return external(2, -1);
case 2: return internal('B', 1);
}
case 'B':
switch (edge) {
case 0: return external(1, +1);
case 1: return internal('A', 2);
case 2: return external(0, 0);
}
case 'V':
switch (edge) {
case 0: return external(1, -1);
case 1: return external(2, +1);
case 2: return internal('A', 0);
}
default:
return fail();
}
case 'U':
switch (child) {
case 'B':
switch (edge) {
case 0: return internal('U', 1);
case 1: return external(2, 0);
case 2: return external(0, +1);
}
case 'U':
switch (edge) {
case 0: return external(1, 0);
case 1: return internal('B', 0);
case 2: return external(0, -1);
}
default:
return fail();
}
case 'V':
switch (child) {
case 'A':
switch (edge) {
case 0: return internal('V', 2);
case 1: return external(0, -1);
case 2: return external(1, 0);
}
case 'V':
switch (edge) {
case 0: return external(2, 0);
case 1: return external(0, +1);
case 2: return internal('A', 0);
}
default:
return fail();
}
case 'C':
switch (child) {
case 'C':
switch (edge) {
case 0: return external(1, +1);
case 1: return internal('Y', 1);
case 2: return external(0, 0);
}
case 'Y':
switch (edge) {
case 0: return external(2, 0);
case 1: return internal('C', 1);
case 2: return external(1, -1);
}
default:
return fail();
}
case 'D':
switch (child) {
case 'D':
switch (edge) {
case 0: return external(2, -1);
case 1: return external(0, 0);
case 2: return internal('X', 2);
}
case 'X':
switch (edge) {
case 0: return external(1, 0);
case 1: return external(2, +1);
case 2: return internal('D', 2);
}
default:
return fail();
}
case 'X':
switch (child) {
case 'C':
switch (edge) {
case 0: return external(2, +1);
case 1: return internal('Y', 1);
case 2: return internal('X', 1);
}
case 'X':
switch (edge) {
case 0: return external(1, 0);
case 1: return internal('C', 2);
case 2: return external(0, -1);
}
case 'Y':
switch (edge) {
case 0: return external(0, +1);
case 1: return internal('C', 1);
case 2: return external(2, -1);
}
default:
return fail();
}
case 'Y':
switch (child) {
case 'D':
switch (edge) {
case 0: return external(1, -1);
case 1: return internal('Y', 2);
case 2: return internal('X', 2);
}
case 'X':
switch (edge) {
case 0: return external(0, -1);
case 1: return external(1, +1);
case 2: return internal('D', 2);
}
case 'Y':
switch (edge) {
case 0: return external(2, 0);
case 1: return external(0, +1);
case 2: return internal('D', 1);
}
default:
return fail();
}
default:
return fail();
}
}
/*
* Compute a transition into a parent triangle, after the above
* function reported an EXTERNAL transition out of a neighbouring
* parent and we had to recurse. Because we're coming inwards, this
* should always return an INTERNAL TransitionResult (or FAIL if it
* gets invalid data).
*/
static TransitionResult transition_in(char parent, unsigned edge, int end)
{
#define EDGEEND(edge, end) (3 * (edge) + 1 + (end))
switch (parent) {
case 'A':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, 0): return internal('A', 1);
case EDGEEND(1, -1): return internal('B', 2);
case EDGEEND(1, +1): return internal('U', 2);
case EDGEEND(2, -1): return internal('U', 0);
case EDGEEND(2, +1): return internal('A', 0);
default:
return fail();
}
case 'B':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, 0): return internal('B', 2);
case EDGEEND(1, -1): return internal('B', 0);
case EDGEEND(1, +1): return internal('V', 0);
case EDGEEND(2, -1): return internal('V', 1);
case EDGEEND(2, +1): return internal('A', 1);
default:
return fail();
}
case 'U':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, -1): return internal('B', 2);
case EDGEEND(0, +1): return internal('U', 2);
case EDGEEND(1, 0): return internal('U', 0);
case EDGEEND(2, 0): return internal('B', 1);
default:
return fail();
}
case 'V':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, -1): return internal('V', 1);
case EDGEEND(0, +1): return internal('A', 1);
case EDGEEND(1, 0): return internal('A', 2);
case EDGEEND(2, 0): return internal('V', 0);
default:
return fail();
}
case 'C':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, 0): return internal('C', 2);
case EDGEEND(1, -1): return internal('C', 0);
case EDGEEND(1, +1): return internal('Y', 2);
case EDGEEND(2, 0): return internal('Y', 0);
default:
return fail();
}
case 'D':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, 0): return internal('D', 1);
case EDGEEND(1, 0): return internal('X', 0);
case EDGEEND(2, -1): return internal('X', 1);
case EDGEEND(2, +1): return internal('D', 0);
default:
return fail();
}
case 'X':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, -1): return internal('Y', 0);
case EDGEEND(0, +1): return internal('X', 2);
case EDGEEND(1, 0): return internal('X', 0);
case EDGEEND(2, -1): return internal('C', 0);
case EDGEEND(2, +1): return internal('Y', 2);
default:
return fail();
}
case 'Y':
switch (EDGEEND(edge, end)) {
case EDGEEND(0, +1): return internal('X', 0);
case EDGEEND(0, -1): return internal('Y', 1);
case EDGEEND(1, -1): return internal('X', 1);
case EDGEEND(1, +1): return internal('D', 0);
case EDGEEND(2, 0): return internal('Y', 0);
default:
return fail();
}
default:
return fail();
}
#undef EDGEEND
}
PenroseCoords *penrose_coords_new(void)
{
PenroseCoords *pc = snew(PenroseCoords);
pc->nc = pc->csize = 0;
pc->c = NULL;
return pc;
}
void penrose_coords_free(PenroseCoords *pc)
{
if (pc) {
sfree(pc->c);
sfree(pc);
}
}
void penrose_coords_make_space(PenroseCoords *pc, size_t size)
{
if (pc->csize < size) {
pc->csize = pc->csize * 5 / 4 + 16;
if (pc->csize < size)
pc->csize = size;
pc->c = sresize(pc->c, pc->csize, char);
}
}
PenroseCoords *penrose_coords_copy(PenroseCoords *pc_in)
{
PenroseCoords *pc_out = penrose_coords_new();
penrose_coords_make_space(pc_out, pc_in->nc);
memcpy(pc_out->c, pc_in->c, pc_in->nc * sizeof(*pc_out->c));
pc_out->nc = pc_in->nc;
return pc_out;
}
/*
* The main recursive function for computing the next triangle's
* combinatorial coordinates.
*/
static void penrosectx_step_recurse(
PenroseContext *ctx, PenroseCoords *pc, size_t depth,
unsigned edge, unsigned *outedge)
{
TransitionResult tr;
penrosectx_extend_coords(ctx, pc, depth+2);
/* Look up the transition out of the starting triangle */
tr = transition(pc->c[depth+1], pc->c[depth], edge);
/* If we've left the parent triangle, recurse to find out what new
* triangle we've landed in at the next size up, and then call
* transition_in to find out which child of that parent we're
* going to */
if (tr.type == EXTERNAL) {
unsigned parent_outedge;
penrosectx_step_recurse(
ctx, pc, depth+1, tr.u.external.parent_edge, &parent_outedge);
tr = transition_in(pc->c[depth+1], parent_outedge, tr.u.external.end);
}
/* Now we should definitely have ended up in a child of the
* (perhaps rewritten) parent triangle */
assert(tr.type == INTERNAL);
pc->c[depth] = tr.u.internal.new_child;
*outedge = tr.u.internal.new_edge;
}
void penrosectx_step(PenroseContext *ctx, PenroseCoords *pc,
unsigned edge, unsigned *outedge)
{
/* Allow outedge to be NULL harmlessly, just in case */
unsigned dummy_outedge;
if (!outedge)
outedge = &dummy_outedge;
penrosectx_step_recurse(ctx, pc, 0, edge, outedge);
}
static Point penrose_triangle_post_edge(char c, unsigned edge)
{
static const Point acute_post_edge[3] = {
{{-1, 1, 0, 1}}, /* phi * t^3 */
{{-1, 1, -1, 1}}, /* t^4 */
{{-1, 1, 0, 0}}, /* 1/phi * t^3 */
};
static const Point obtuse_post_edge[3] = {
{{0, -1, 1, 0}}, /* 1/phi * t^4 */
{{0, 0, 1, 0}}, /* t^2 */
{{-1, 0, 0, 1}}, /* phi * t^4 */
};
switch (c) {
case 'A': case 'B': case 'C': case 'D':
return acute_post_edge[edge];
default: /* case 'U': case 'V': case 'X': case 'Y': */
return obtuse_post_edge[edge];
}
}
void penrose_place(PenroseTriangle *tri, Point u, Point v, int index_of_u)
{
unsigned i;
Point here = u, delta = point_sub(v, u);
for (i = 0; i < 3; i++) {
unsigned edge = (index_of_u + i) % 3;
tri->vertices[edge] = here;
here = point_add(here, delta);
delta = point_mul(delta, penrose_triangle_post_edge(
tri->pc->c[0], edge));
}
}
void penrose_free(PenroseTriangle *tri)
{
penrose_coords_free(tri->pc);
sfree(tri);
}
static bool penrose_relative_probability(char c)
{
/* Penrose tile probability ratios are always phi, so we can
* approximate that very well using two consecutive Fibonacci
* numbers */
switch (c) {
case 'A': case 'B': case 'X': case 'Y':
return 165580141;
case 'C': case 'D': case 'U': case 'V':
return 102334155;
default:
return 0;
}
}
static char penrose_choose_random(const char *possibilities, random_state *rs)
{
const char *p;
unsigned long value, limit = 0;
for (p = possibilities; *p; p++)
limit += penrose_relative_probability(*p);
value = random_upto(rs, limit);
for (p = possibilities; *p; p++) {
unsigned long curr = penrose_relative_probability(*p);
if (value < curr)
return *p;
value -= curr;
}
assert(false && "Probability overflow!");
return possibilities[0];
}
static const char *penrose_starting_tiles(int which)
{
return which == PENROSE_P2 ? "ABUV" : "CDXY";
}
static const char *penrose_valid_parents(char tile)
{
switch (tile) {
case 'A': return "ABV";
case 'B': return "ABU";
case 'U': return "AU";
case 'V': return "BV";
case 'C': return "CX";
case 'D': return "DY";
case 'X': return "DXY";
case 'Y': return "CXY";
default: return NULL;
}
}
void penrosectx_init_random(PenroseContext *ctx, random_state *rs, int which)
{
ctx->rs = rs;
ctx->must_free_rs = false;
ctx->prototype = penrose_coords_new();
penrose_coords_make_space(ctx->prototype, 1);
ctx->prototype->c[0] = penrose_choose_random(
penrose_starting_tiles(which), rs);
ctx->prototype->nc = 1;
ctx->start_vertex = random_upto(rs, 3);
ctx->orientation = random_upto(rs, 10);
}
void penrosectx_init_from_params(
PenroseContext *ctx, const struct PenrosePatchParams *ps)
{
size_t i;
ctx->rs = NULL;
ctx->must_free_rs = false;
ctx->prototype = penrose_coords_new();
penrose_coords_make_space(ctx->prototype, ps->ncoords);
for (i = 0; i < ps->ncoords; i++)
ctx->prototype->c[i] = ps->coords[i];
ctx->prototype->nc = ps->ncoords;
ctx->start_vertex = ps->start_vertex;
ctx->orientation = ps->orientation;
}
void penrosectx_cleanup(PenroseContext *ctx)
{
if (ctx->must_free_rs)
random_free(ctx->rs);
penrose_coords_free(ctx->prototype);
}
PenroseCoords *penrosectx_initial_coords(PenroseContext *ctx)
{
return penrose_coords_copy(ctx->prototype);
}
void penrosectx_extend_coords(PenroseContext *ctx, PenroseCoords *pc,
size_t n)
{
if (ctx->prototype->nc < n) {
penrose_coords_make_space(ctx->prototype, n);
while (ctx->prototype->nc < n) {
if (!ctx->rs) {
/*
* For safety, similarly to spectre.c, we respond to a
* lack of available random_state by making a
* deterministic one.
*/
ctx->rs = random_new("dummy", 5);
ctx->must_free_rs = true;
}
ctx->prototype->c[ctx->prototype->nc] = penrose_choose_random(
penrose_valid_parents(ctx->prototype->c[ctx->prototype->nc-1]),
ctx->rs);
ctx->prototype->nc++;
}
}
penrose_coords_make_space(pc, n);
while (pc->nc < n) {
pc->c[pc->nc] = ctx->prototype->c[pc->nc];
pc->nc++;
}
}
static Point penrose_triangle_edge_0_length(char c)
{
static const Point one = {{ 1, 0, 0, 0 }};
static const Point phi = {{ 1, 0, 1, -1 }};
static const Point invphi = {{ 0, 0, 1, -1 }};
switch (c) {
/* P2 tiling: unit-length edges are the long edges, i.e. edges
* 1,2 of AB and edge 0 of UV. So AB have edge 0 short. */
case 'A': case 'B':
return invphi;
case 'U': case 'V':
return one;
/* P3 tiling: unit-length edges are edges 1,2 of everything,
* so CD have edge 0 short and XY have it long. */
case 'C': case 'D':
return invphi;
default: /* case 'X': case 'Y': */
return phi;
}
}
PenroseTriangle *penrose_initial(PenroseContext *ctx)
{
char type = ctx->prototype->c[0];
Point origin = {{ 0, 0, 0, 0 }};
Point edge0 = penrose_triangle_edge_0_length(type);
Point negoffset;
size_t i;
PenroseTriangle *tri = snew(PenroseTriangle);
/* Orient the triangle by deciding what vector edge #0 should traverse */
edge0 = point_mul(edge0, point_rot(ctx->orientation));
/* Place the triangle at an arbitrary position, in that orientation */
tri->pc = penrose_coords_copy(ctx->prototype);
penrose_place(tri, origin, edge0, 0);
/* Now translate so that the appropriate vertex is at the origin */
negoffset = tri->vertices[ctx->start_vertex];
for (i = 0; i < 3; i++)
tri->vertices[i] = point_sub(tri->vertices[i], negoffset);
return tri;
}
PenroseTriangle *penrose_adjacent(PenroseContext *ctx,
const PenroseTriangle *src_tri,
unsigned src_edge, unsigned *dst_edge_out)
{
unsigned dst_edge;
PenroseTriangle *dst_tri = snew(PenroseTriangle);
dst_tri->pc = penrose_coords_copy(src_tri->pc);
penrosectx_step(ctx, dst_tri->pc, src_edge, &dst_edge);
penrose_place(dst_tri, src_tri->vertices[(src_edge+1) % 3],
src_tri->vertices[src_edge], dst_edge);
if (dst_edge_out)
*dst_edge_out = dst_edge;
return dst_tri;
}
static int penrose_cmp(void *av, void *bv)
{
PenroseTriangle *a = (PenroseTriangle *)av, *b = (PenroseTriangle *)bv;
size_t i, j;
/* We should only ever need to compare the first two vertices of
* any triangle, because those force the rest */
for (i = 0; i < 2; i++) {
for (j = 0; j < 4; j++) {
int ac = a->vertices[i].coeffs[j], bc = b->vertices[i].coeffs[j];
if (ac < bc)
return -1;
if (ac > bc)
return +1;
}
}
return 0;
}
static unsigned penrose_sibling_edge_index(char c)
{
switch (c) {
case 'A': case 'U': return 2;
case 'B': case 'V': return 1;
default: return 0;
}
}
void penrosectx_generate(
PenroseContext *ctx,
bool (*inbounds)(void *inboundsctx,
const PenroseTriangle *tri), void *inboundsctx,
void (*tile)(void *tilectx, const Point *vertices), void *tilectx)
{
tree234 *placed = newtree234(penrose_cmp);
PenroseTriangle *qhead = NULL, *qtail = NULL;
{
PenroseTriangle *tri = penrose_initial(ctx);
add234(placed, tri);
tri->next = NULL;
tri->reported = false;
if (inbounds(inboundsctx, tri))
qhead = qtail = tri;
}
while (qhead) {
PenroseTriangle *tri = qhead;
unsigned edge;
unsigned sibling_edge = penrose_sibling_edge_index(tri->pc->c[0]);
for (edge = 0; edge < 3; edge++) {
PenroseTriangle *new_tri, *found_tri;
new_tri = penrose_adjacent(ctx, tri, edge, NULL);
if (!inbounds(inboundsctx, new_tri)) {
penrose_free(new_tri);
continue;
}
found_tri = find234(placed, new_tri, NULL);
if (found_tri) {
if (edge == sibling_edge && !tri->reported &&
!found_tri->reported) {
/*
* found_tri and tri are opposite halves of the
* same tile; both are in the tree, and haven't
* yet been reported as a completed tile.
*/
unsigned new_sibling_edge = penrose_sibling_edge_index(
found_tri->pc->c[0]);
Point tilevertices[4] = {
tri->vertices[(sibling_edge + 1) % 3],
tri->vertices[(sibling_edge + 2) % 3],
found_tri->vertices[(new_sibling_edge + 1) % 3],
found_tri->vertices[(new_sibling_edge + 2) % 3],
};
tile(tilectx, tilevertices);
tri->reported = true;
found_tri->reported = true;
}
penrose_free(new_tri);
continue;
}
add234(placed, new_tri);
qtail->next = new_tri;
qtail = new_tri;
new_tri->next = NULL;
new_tri->reported = false;
}
qhead = qhead->next;
}
{
PenroseTriangle *tri;
while ((tri = delpos234(placed, 0)) != NULL)
penrose_free(tri);
freetree234(placed);
}
}
const char *penrose_tiling_params_invalid(
const struct PenrosePatchParams *params, int which)
{
size_t i;
if (params->ncoords == 0)
return "expected at least one coordinate";
for (i = 0; i < params->ncoords; i++) {
if (!penrose_valid_letter(params->coords[i], which))
return "invalid coordinate letter";
if (i > 0 && !strchr(penrose_valid_parents(params->coords[i-1]),
params->coords[i]))
return "invalid pair of consecutive coordinates";
}
return NULL;
}
struct PenroseOutputCtx {
int xoff, yoff;
Coord xmin, xmax, ymin, ymax;
penrose_tile_callback_fn external_cb;
void *external_cbctx;
};
static bool inbounds(void *vctx, const PenroseTriangle *tri)
{
struct PenroseOutputCtx *octx = (struct PenroseOutputCtx *)vctx;
size_t i;
for (i = 0; i < 3; i++) {
Coord x = point_x(tri->vertices[i]);
Coord y = point_y(tri->vertices[i]);
if (coord_cmp(x, octx->xmin) < 0 || coord_cmp(x, octx->xmax) > 0 ||
coord_cmp(y, octx->ymin) < 0 || coord_cmp(y, octx->ymax) > 0)
return false;
}
return true;
}
static void null_output_tile(void *vctx, const Point *vertices)
{
}
static void really_output_tile(void *vctx, const Point *vertices)
{
struct PenroseOutputCtx *octx = (struct PenroseOutputCtx *)vctx;
size_t i;
int coords[16];
for (i = 0; i < 4; i++) {
Coord x = point_x(vertices[i]);
Coord y = point_y(vertices[i]);
coords[4*i + 0] = x.c1 + octx->xoff;
coords[4*i + 1] = x.cr5;
coords[4*i + 2] = y.c1 + octx->yoff;
coords[4*i + 3] = y.cr5;
}
octx->external_cb(octx->external_cbctx, coords);
}
static void penrose_set_bounds(struct PenroseOutputCtx *octx, int w, int h)
{
octx->xoff = w/2;
octx->yoff = h/2;
octx->xmin.c1 = -octx->xoff;
octx->xmax.c1 = -octx->xoff + w;
octx->ymin.c1 = octx->yoff - h;
octx->ymax.c1 = octx->yoff;
octx->xmin.cr5 = 0;
octx->xmax.cr5 = 0;
octx->ymin.cr5 = 0;
octx->ymax.cr5 = 0;
}
void penrose_tiling_randomise(struct PenrosePatchParams *params, int which,
int w, int h, random_state *rs)
{
PenroseContext ctx[1];
struct PenroseOutputCtx octx[1];
penrose_set_bounds(octx, w, h);
penrosectx_init_random(ctx, rs, which);
penrosectx_generate(ctx, inbounds, octx, null_output_tile, NULL);
params->orientation = ctx->orientation;
params->start_vertex = ctx->start_vertex;
params->ncoords = ctx->prototype->nc;
params->coords = snewn(params->ncoords, char);
memcpy(params->coords, ctx->prototype->c, params->ncoords);
penrosectx_cleanup(ctx);
}
void penrose_tiling_generate(
const struct PenrosePatchParams *params, int w, int h,
penrose_tile_callback_fn cb, void *cbctx)
{
PenroseContext ctx[1];
struct PenroseOutputCtx octx[1];
penrose_set_bounds(octx, w, h);
octx->external_cb = cb;
octx->external_cbctx = cbctx;
penrosectx_init_from_params(ctx, params);
penrosectx_generate(ctx, inbounds, octx, really_output_tile, octx);
penrosectx_cleanup(ctx);
}