# triangle_intersection module

This is a collection of procedures and types for computing intersections between two Bézier triangles in $$\mathbf{R}^2$$. The region(s) of intersection (if non-empty) will be curved polygons defined by the Bézier curve segments that form the exterior.

Note

In most of the procedures below both the number of nodes $$N$$ and the degree $$d$$ of a Bézier triangle are provided. It is redundant to require both as arguments since $$N = \binom{d + 2}{2}$$. However, both are provided as arguments to avoid unnecessary re-computation, i.e. we expect the caller to know both $$N$$ and $$d$$.

## Procedures

void BEZ_locate_point_triangle(const int *num_nodes, const double *nodes, const int *degree, const double *x_val, const double *y_val, double *s_val, double *t_val)

This solves the inverse problem $$B(s, t) = (x, y)$$ (if it can be solved). Does so by subdividing the triangle until the sub-triangles are sufficiently small, then using Newton’s method to narrow in on the pre-image of the point.

This assumes the triangle is “valid”, i.e. it has positive Jacobian throughout the unit triangle. (If this were not true, then multiple inputs could map to the same output.)

Parameters:
• num_nodes (const int*) – [Input] The number of nodes $$N$$ in the control net of the Bézier triangle.

• nodes (const double*) – [Input] The actual control net of the Bézier triangle as a $$2 \times N$$ array. This should be laid out in Fortran order, with $$2 N$$ total values.

• degree (const int*) – [Input] The degree $$d$$ of the Bézier triangle.

• x_val (const double*) – [Input] The $$x$$-value of the point being located.

• y_val (const double*) – [Input] The $$y$$-value of the point being located.

• s_val (double*) – [Output] The first parameter $$s$$ of the solution. If $$(x, y)$$ can’t be located on the triangle, then s_val = -1.0.

• t_val (double*) – [Output] The second parameter $$t$$ of the solution. If $$(x, y)$$ can’t be located on the triangle, then this value is undefined.

Signature:

void
BEZ_locate_point_triangle(const int *num_nodes,
const double *nodes,
const int *degree,
const double *x_val,
const double *y_val,
double *s_val,
double *t_val);

void BEZ_newton_refine_triangle(const int *num_nodes, const double *nodes, const int *degree, const double *x_val, const double *y_val, const double *s, const double *t, double *updated_s, double *updated_t)

This refines a solution to $$B(s, t) = (x, y) = p$$ using Newton’s method. Given a current approximation $$(s_n, t_n)$$ for a solution, this produces the updated approximation via

$\begin{split}\left[\begin{array}{c} s_{n + 1} \\ t_{n + 1} \end{array}\right] = \left[\begin{array}{c} s_n \\ t_n \end{array}\right] - DB(s_n, t_n)^{-1} \left(B(s_n, t_n) - p\right).\end{split}$
Parameters:
• num_nodes (const int*) – [Input] The number of nodes $$N$$ in the control net of the Bézier triangle.

• nodes (const double*) – [Input] The actual control net of the Bézier triangle as a $$2 \times N$$ array. This should be laid out in Fortran order, with $$2 N$$ total values.

• degree (const int*) – [Input] The degree $$d$$ of the Bézier triangle.

• x_val (const double*) – [Input] The $$x$$-value of the point $$p$$.

• y_val (const double*) – [Input] The $$y$$-value of the point $$p$$.

• s (const double*) – [Input] The first parameter $$s_n$$ of the current approximation of a solution.

• t (const double*) – [Input] The second parameter $$t_n$$ of the current approximation of a solution.

• updated_s (double*) – [Output] The first parameter $$s_{n + 1}$$ of the updated approximation.

• updated_t (double*) – [Output] The second parameter $$t_{n + 1}$$ of the updated approximation.

Signature:

void
BEZ_newton_refine_triangle(const int *num_nodes,
const double *nodes,
const int *degree,
const double *x_val,
const double *y_val,
const double *s,
const double *t,
double *updated_s,
double *updated_t);

void BEZ_triangle_intersections(const int *num_nodes1, const double *nodes1, const int *degree1, const int *num_nodes2, const double *nodes2, const int *degree2, const int *segment_ends_size, int *segment_ends, const int *segments_size, CurvedPolygonSegment *segments, int *num_intersected, TriangleContained *contained, Status *status)

Compute the intersection of two Bézier triangles. This will first compute all intersection points between edges of the first and second triangle (nine edge pairs in total). Then, it will classify each point according to which triangle is “interior” at that point. Finally, it will form a loop of intersection points using the classifications until all intersections have been used or discarded.

Tip

If the status returned is INSUFFICIENT_SPACE that means either

• segment_ends_size is smaller than num_intersected so segment_ends needs to be resized to at least as large as num_intersected.

• segments_size is smaller than the number of segments. The number of segments will be the last index in the list of edge indices: segment_ends[num_intersected - 1]. In this case segments needs to be resized.

This means a successful invocation of this procedure may take three attempts. To avoid false starts occurring on a regular basis, keep a static workspace around that will continue to grow as resizing is needed, but will never shrink.

Parameters:
• num_nodes1 (const int*) – [Input] The number of nodes $$N_1$$ in the control net of the first Bézier triangle.

• nodes1 (const double*) – [Input] The actual control net of the first Bézier triangle as a $$2 \times N_1$$ array. This should be laid out in Fortran order, with $$2 N_1$$ total values.

• degree1 (const int*) – [Input] The degree $$d_1$$ of the first Bézier triangle.

• num_nodes2 (const int*) – [Input] The number of nodes $$N_2$$ in the control net of the second Bézier triangle.

• nodes2 (const double*) – [Input] The actual control net of the second Bézier triangle as a $$2 \times N_2$$ array. This should be laid out in Fortran order, with $$2 N_2$$ total values.

• degree2 (const int*) – [Input] The degree $$d_2$$ of the second Bézier triangle.

• segment_ends_size (const int*) – [Input] The size of segment_ends, which must be pre-allocated by the caller.

• segment_ends (int*) – [Output] An array (pre-allocated by the caller) of the end indices for each group of segments in segments. For example, if the triangles intersect in two distinct curved polygons, the first of which has four sides and the second of which has three, then the first two values in segment_ends will be [4, 7] and num_intersected will be 2.

• segments_size (const int*) – [Input] The size of segments, which must be pre-allocated by the caller.

• segments () – [Output] An array (pre-allocated by the caller) of the edge segments that make up the boundary of the curved polygon(s) that form the intersection of the two triangles.

• num_intersected (int*) – [Output] The number of curved polygons in the intersection of two triangles.

• contained () – [Output] Enum indicating if one triangle is fully contained in the other.

• status () –

[Output] The status code for the procedure. Will be

• SUCCESS on success.

• INSUFFICIENT_SPACE if segment_ends_size is smaller than num_intersected or if segments_size is smaller than the number of segments.

• UNKNOWN if the intersection points are classified in an unexpected way (e.g. if there is both an ignored corner and a tangent intersection, but no other types).

• NO_CONVERGE if the two curves in an edge pair don’t converge to approximately linear after being subdivided 20 times. (This error will occur via BEZ_curve_intersections().)

• An integer $$N_C \geq 64$$ to indicate that there were $$N_C$$ pairs of candidate segments during edge-edge intersection that had overlapping convex hulls. This is a sign of either round-off error in detecting that the edges are coincident curve segments on the same algebraic curve or that the intersection is a non-simple root. (This error will occur via BEZ_curve_intersections().)

• BAD_MULTIPLICITY if the two curves in an edge pair have an intersection that doesn’t converge to either a simple or double root via Newton’s method. (This error will occur via BEZ_curve_intersections().)

• EDGE_END If there is an attempt to add an intersection point with either the $$s$$ or $$t$$-parameter equal to 1 (i.e. if the intersection is at the end of an edge). This should not occur because such intersections are “rotated” to the beginning of the neighboring edge before the boundary of the curved polygon is formed.

• SAME_CURVATURE if the two curves in an edge pair have identical curvature at a tangent intersection.

• BAD_INTERIOR if a curved polygon requires more than 10 sides. This could be due to either a particular complex intersection, a programming error or round-off which causes an infinite loop of intersection points to be added without wrapping around back to the first intersection point.

Signature:

void
BEZ_triangle_intersections(const int *num_nodes1,
const double *nodes1,
const int *degree1,
const int *num_nodes2,
const double *nodes2,
const int *degree2,
const int *segment_ends_size,
int *segment_ends,
const int *segments_size,
CurvedPolygonSegment *segments,
int *num_intersected,
TriangleContained *contained,
Status *status);

void BEZ_free_triangle_intersections_workspace(void)

This frees any long-lived workspace(s) used by libbezier throughout the life of a program. It should be called during clean-up for any code which invokes BEZ_triangle_intersections().

Signature:

void
BEZ_free_triangle_intersections_workspace(void);


## Types

struct CurvedPolygonSegment

Describes an edge of a CurvedPolygon formed when intersecting two curved Bézier triangles. The edges of the intersection need not be an entire edge of one of the triangles. For example, an edge $$E(s)$$ may be restricted to $$E\left(\left[\frac{1}{4}, \frac{7}{8}\right]\right)$$.

double start

The start parameter of the segment. In the restriction $$E\left(\left[\frac{1}{4}, \frac{7}{8}\right]\right)$$, the start would be 0.25.

double end

The end parameter of the segment. In the restriction $$E\left(\left[\frac{1}{4}, \frac{7}{8}\right]\right)$$, the end would be 0.875.

int edge_index

An index describing which edge the segment falls on. The edges of the first triangle in the intersection are given index values of 1, 2 and 3 while those of the second triangle are 4, 5 and 6.

In the header bezier/triangle_intersection.h, this is defined as

typedef struct CurvedPolygonSegment {
double start;
double end;
int edge_index;
} CurvedPolygonSegment;

enum TriangleContained

This enum is used to indicate if one triangle is contained in another when doing triangle-triangle intersection.

enumerator NEITHER

(0) Indicates that neither triangle is contained in the other. This could mean the triangles are disjoint or that they intersect in a way other than full containment.

enumerator FIRST

(1) Indicates that the first triangle (arguments will be ordered) is fully contained in the second. This allows for points of tangency, shared corners or shared segments along an edge.

enumerator SECOND

(2) Indicates that the second triangle (arguments will be ordered) is fully contained in the first. This allows for points of tangency, shared corners or shared segments along an edge.