5 Geometry

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5 Geometry

(require sfont/geometry)

package: sfont

This is the module for vector operations, transformations and bezier paths.

value
gen:geometric : any/c

Generic interface that defines methods for geometric transformations.

parameter
(precision) → real?
(precision p) → void?
p : real?

A parameter that controls rounding.

procedure
(approx n) → real?
n : real?

Approximate the numbers with the precision defined by precision.

struct
(struct vec (x y))
x : real?
y : real?

A structure that represents a 2D vector. A vec implements the gen:geometric interface.

struct
(struct vec3 (x y z))
  x : real?
  y : real?
  z : real?

A structure that represents a 3D vector.

procedure
(vec3->vec v) → vec?
v : vec3?

procedure
(vec->vec3 v) → vec3?
v : vec?

Conversion between 3D and 2D vcectors.

struct
(struct trans-mat (x xy yx y x-offset y-offset))
  x : real?
  xy : real?
  yx : real?
  y : real?
  x-offset : real?
  y-offset : real?

A transformation matrix. A trans-mat implements the gen:geometric interface.

procedure
(geometric? o) → boolean?
o : any/c

A predicate for structures that implement the generic group gen:geometric

procedure
(transform o m) → geometric?
o : geometric?
m : trans-mat?

Applies the trasformation matrix to the object.

procedure
(translate o x y) → geometric?
  o : geometric?
  x : real?
  y : real?

Applies the translation to the object.

procedure
(rotate o angle) → geometric?
o : geometric?
angle : real?

Applies the rotation to the object.

procedure
(scale o fx [fy]) → geometric?
  o : geometric?
  fx : real?
  fy : real? = fx

Scales the object.

procedure
(skew-x o angle) → geometric?
o : geometric?
angle : real?

Applies a shear transformation to the object.

procedure
(skew-y o angle) → geometric?
o : geometric?
angle : real?

Applies a shear transformation to the object.

procedure
(reflect-x o) → geometric?
o : geometric?

Reflects the object around the x axis.

procedure
(reflect-y o) → geometric?
o : geometric?

Reflects the object around the y axis.

procedure
(vec= v1 v2) → boolean?
v1 : vec?
v2 : vec?

True if the x and y coordinates of the vectors are equal.

procedure
(vec-approx= v1 v2) → boolean?
v1 : vec?
v2 : vec?

True if the x and y coordinates of the vectors are approximately equal (using the precision parameter).

procedure
(list->vec l) → vec?
l : (list/c real? real?)

Produces a vec from a list of two numbers.

procedure
(vec->list v) → (list/c real? real?)
v : vec?

Produces a list of two numbers from a vec.

procedure
(vec-length v) → (and/c real? (not/c negative?))
v : vec?

Produces the length of the vector.

procedure
(vec-angle v) → real?
v : vec?

Produces the angle of the vector.

procedure
(vec+ v1 v2) → vec?
v1 : vec?
v2 : vec?

Sum two vectors.

procedure
(vec- v1 v2) → vec?
v1 : vec?
v2 : vec?

Subtract two vectors.

procedure
(vec* v1 n) → vec?
v1 : vec?
n : real?

Scalar multiplications.

procedure
(vec/ v1 n) → vec?
v1 : vec?
n : real?

Scalar division.

procedure
(aligned? v1 v2 v3) → boolean?
  v1 : vec?
  v2 : vec?
  v3 : vec?

True if the vectors v1, v1 and v1 are aligned.

procedure
(translation-matrix x y) → trans-mat?
x : real?
y : real?

Produces a translation matrix.

procedure
(rotation-matrix angle) → trans-mat?
angle : real?

Produces a rotation matrix.

procedure
(scale-matrix fx [fy]) → trans-mat?
fx : real?
fy : real? = fx

Produces a scale matrix.

procedure
(shear-matrix x y) → trans-mat?
x : real?
y : real?

Produces a shear matrix.

procedure
(trans-mat* m1 m2) → trans-mat?
m1 : trans-mat?
m2 : trans-mat?

Multiplies two transformation matrices.

procedure
(trans-mat-vec* m v) → vec3?
m : trans-mat?
v : vec3?

Multiplies the translation matrix and the 3D vector.

procedure
(dot-prod v1 v2) → real?
v1 : vec?
v2 : vec?

Produces the dot product of two 2D vectors.

procedure
(dot-prod-3 v1 v2) → real?
v1 : vec3?
v2 : vec3?

Produces the dot product of two 3D vectors.

procedure
(cross-prod-2d v1 v2) → real?
v1 : vec?
v2 : vec?

The third scalar component of the cross product.

procedure
(segment-intersection v1 v2 v3 v4) → (or/c vec? #f)
  v1 : vec?
  v2 : vec?
  v3 : vec?
  v4 : vec?

The intersection (if any) between the segments v1-v2 and v3-v4.

procedure
(signed-area v1 v2) → real?
v1 : vec?
v2 : vec?

The signed area of the triangle formed by the two vectors. Positive if the angle between vectors is counter-clockwise.

procedure
(signed-polygonal-area lov) → real?
lov : (listof vec?)

The signed area of the polygon formed by the list of vectors. Positive if counter-clockwise.

procedure
(intersect-hor h v1 v2) → (or/c vec? #f)
  h : real?
  v1 : vec?
  v2 : vec?

Produces the intersection (if any) between the horizontal line passing from (0, h) and the line segment v1-v2.

procedure
(intersect-vert v v1 v2) → (or/c vec? #f)
  v : real?
  v1 : vec?
  v2 : vec?

Produces the intersection (if any) between the vertical line passing from (v, 0) and the line segment v1-v2.

procedure
(pass-through-hor? h v1 v2) → boolean?
  h : real?
  v1 : vec?
  v2 : vec?

True if the line segment v1-v2 intersects horizontal line passing from (0, h).

procedure
(pass-through-vert? v v1 v2) → boolean?
  v : real?
  v1 : vec?
  v2 : vec?

True if the line segment v1-v2 intersects the vertical line passing from (v, 0).

5.1 Bezier Curves

In Sfont a ’bezier path’ is a list of vec, a segment is a bezier path with only two ’on-curve’ points.

value
bezier/c : flat-contract?

value
segment/c : flat-contract?

value
cubic-bezier/c : flat-contract?

value
cubic-segment/c : flat-contract?

value
closed-bezier/c : flat-contract?

procedure
(closed? b) → boolean?
b : bezier/c

True if the first and last points coincide.

procedure
(segments b [o]) → (listof segment/c)
b : bezier/c
o : natural-number/c = 3

’Explodes’ a path in a list of segments. The optional arguments declares the ’order’ of the bezier path (basically: the number of control points for each segments plus one, cubic beziers have two points, quadratic one, etc.

procedure
(on-curve-points b [o]) → (listof vec?)
b : bezier/c
o : natural-number/c = 3

Produce a list of points removing all the control points from the bezier.

procedure
(off-curve-points b [o]) → (listof vec?)
b : bezier/c
o : natural-number/c = 3

Produce a list of control points of the bezier path.

procedure
(end-points b) → (cons/c vec? vec?)
b : bezier/c

Produce a pair with the first and last point of the path.

procedure
(line-segment? s) → boolean?
s : segment/c

True if the segment represents a line (ie. the points are aligned)

procedure
(canonical-line-segment s) → cubic-segment/c
s : cubic-segment/c

Produces a cubic line segment where control points are placed at the extrema.

procedure
(split s t) →
segment/c segment/c
s : segment/c
t : (real-in 0 1)

Splits the segment in two parts. If t is 0 or 1 the first/second half is an empty list.

procedure
(split-at-point s v) →
segment/c segment/c
s : segment/c
v : vec?

Like split but try to split the path near the point provided.

procedure
(join-beziers b1 b ...) → bezier/c
b1 : bezier/c
b : bezier/c

Concatenate bezier paths. If the last and first points of two consecutive paths are not vec-approx= an error is raised.

procedure
(point-at s t) → vec?
s : segment/c
t : (real-in 0 1)

Find the point on a segment at ’time’ t.

procedure
(polygonize-segment s n) → (listof vec?)
s : segment/c
n : natural-number/c

Transform the segment in a polygon with n sides.

procedure
(end-points-bounding-box s) → bounding-box/c
s : segment/c

Produces the bounding box of the line segment that joins the endpoints of the bezier segment.

procedure
(end-points-at-extrema? s) → boolean?
s : segment/c

True if the bounding-box of the segment and end-points-bounding-box are the same.

procedure
(segment-bounding-box s) → bounding-box/c
s : segment/c

Produces the bounding-box of the segment.

procedure
(bezier-bounding-box b [o]) → bounding-box/c
b : bezier/c
o : natural-number/c = 3

Produces the bounding-box of the bezier path.

procedure
(bezier-signed-area b [o s]) → real?
  b : bezier/c
  o : natural-number/c = 3
  s : natural-number/c = 200

The area of the bezier path of order o. The path will be trasformed in a polygon, s controls in how many sides every segment is divided. Positive if the path is counter-clockwise.

procedure
(bezier-area b [o s]) → (and/c real? positive?)
  b : bezier/c
  o : natural-number/c = 3
  s : natural-number/c = 200

The absolute value of bezier-signed-area.

procedure
(clockwise? b) → boolean?
b : bezier/c

True if the path is clockwise.

procedure
(clockwise b) → bezier/c
b : bezier/c

Reverses the path if it isn’t clockwise.

procedure
(cubic-bezier-intersections b1 b2) → (listof vec?)
b1 : cubic-bezier/c
b2 : cubic-bezier/c

Produces a list of intersections between the two cubic bezier paths.

procedure
(cubic-segments-intersections s1 s2) → (listof vec?)
s1 : cubic-segment/c
s2 : cubic-segment/c

Produces a list of intersections between the two cubic bezier segments.

procedure
(line-segment-intersections l s) → (listof vec?)
l : segment/c
s : segment/c

Produces a list of intersections between the line segment and the segment.

procedure
(segment-intersect-hor h s) → (listof vec?)
h : real?
s : segment/c

Produces a list of intersections between the bezier segment and a the horizontal line y = h.

procedure
(segment-intersect-vert v s) → (listof vec?)
v : real?
s : segment/c

Produces a list of intersections between the bezier segment and a the vertical line x = v.

procedure
(bezier-intersect-hor h s) → (listof vec?)
h : real?
s : segment/c

Produces a list of intersections between the bezier path and a the horizontal line y = h.

procedure
(bezier-intersect-vert v s) → (listof vec?)
v : real?
s : segment/c

Produces a list of intersections between the bezier path and a the vertical line x = v.

procedure
(bezier-boundaries-hor h s) → bounding-box/c
h : real?
s : segment/c

Produces the bounding box of the intersections between the bezier path and a the horizontal line y = h.

procedure
(point-inside-bezier? v b) → boolean?
v : vec?
b : closed-bezier/c

True if the point is inside the bezier path.

procedure
(bezier->path b path) → (is-a?/c dc-path%)
b : cubic-bezier/c
path : (is-a?/c dc-path%)

Write the bezier to a dc-path%.

procedure
(print-beziers b ...) → pict?
b : cubic-bezier/c

Print a graphic representation of the cubic bezier paths.

5.1.1 Boolean operations

procedure
(bezier-subtract b1 b2) → (listof closed-bezier/c)
b1 : closed-bezier/c
b2 : closed-bezier/c

Subtract the second closed bezier path from the first, produces a list of closed bezier paths.

procedure
(bezier-union b1 b2) → (listof closed-bezier/c)
b1 : closed-bezier/c
b2 : closed-bezier/c

Add the second closed bezier path to the first, produces a list of closed bezier paths (if the paths don’t intersect, for example, the two original paths are returned).

procedure
(bezier-intersection b1 b2) → (listof closed-bezier/c)
b1 : closed-bezier/c
b2 : closed-bezier/c

Intersect the second closed bezier path with the first, produces a list of closed bezier paths.

5.2 Bounding Boxes

value
bounding-box/c : flat-contract?

A Bounding Box can be a pair of vec with the first one representing the lower left corner and the second one the upper right corner, or #f for the null Bounding Box.

procedure
(combine-bounding-boxes bb b ...) → bounding-box/c
bb : bounding-box/c
b : bounding-box/c

Produces the bounding box of the bounding boxes.

procedure
(line-bounding-box l) → bounding-box/c
l : (cons/c vec? vec?)

Produces the bounding box of the line segment represented by the two vec.

procedure
(inside-bounding-box? v bb) → boolean?
v : vec?
bb : bounding-box/c

True if the vec is inside the bounding box.

procedure
(overlap-bounding-boxes? bb1 bb2) → boolean?
bb1 : bounding-box/c
bb2 : bounding-box/c

True if the bounding boxes overlap.

procedure
(include-bounding-box? bb1 bb2) → boolean?
bb1 : bounding-box/c
bb2 : bounding-box/c

True if the second bounding box is surrounded by the first one.

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1	Font structures and contracts
2	Spacing fonts
3	Accessing and modifying data
4	Interpolations and font math
5	Geometry
6	Parametric fonts and font macros
7	Graphical interfaces
8	Utilities