Next: Pens, Previous: Data types, Up: Programming [Contents][Index]
path
a cubic spline resolved into a fixed path.
The implicit initializer for paths is nullpath
.
For example, the routine circle(pair c, real r)
, which returns a
Bezier curve approximating a circle of radius r
centered on c
,
is based on unitcircle
(see unitcircle):
path circle(pair c, real r) { return shift(c)*scale(r)*unitcircle; }
If high accuracy is needed, a true circle may be produced with the
routine Circle
defined in the module graph.asy
:
import graph; path Circle(pair c, real r, int n=nCircle);
A circular arc consistent with circle
centered on
c
with radius r
from angle1
to angle2
degrees, drawing counterclockwise if angle2 >= angle1
, can be
constructed with
path arc(pair c, real r, real angle1, real angle2);
One may also specify the direction explicitly:
path arc(pair c, real r, real angle1, real angle2, bool direction);
Here the direction can be specified as CCW (counter-clockwise) or CW
(clockwise). For convenience, an arc centered at c
from pair
z1
to z2
(assuming |z2-c|=|z1-c|
) in the may also
be constructed with
path arc(pair c, explicit pair z1, explicit pair z2, bool direction=CCW)
If high accuracy is needed, true arcs may be produced with routines
in the module graph.asy
that produce Bezier curves with n
control points:
import graph; path Arc(pair c, real r, real angle1, real angle2, bool direction, int n=nCircle); path Arc(pair c, real r, real angle1, real angle2, int n=nCircle); path Arc(pair c, explicit pair z1, explicit pair z2, bool direction=CCW, int n=nCircle);
An ellipse can be drawn with the routine
path ellipse(pair c, real a, real b) { return shift(c)*scale(a,b)*unitcircle; }
A brace can be constructed between pairs a
and b
with
path brace(pair a, pair b, real amplitude=bracedefaultratio*length(b-a));
This example illustrates the use of all five guide connectors discussed in Tutorial and Bezier curves:
size(300,0); pair[] z=new pair[10]; z[0]=(0,100); z[1]=(50,0); z[2]=(180,0); for(int n=3; n <= 9; ++n) z[n]=z[n-3]+(200,0); path p=z[0]..z[1]---z[2]::{up}z[3] &z[3]..z[4]--z[5]::{up}z[6] &z[6]::z[7]---z[8]..{up}z[9]; draw(p,grey+linewidth(4mm)); dot(z);
Here are some useful functions for paths:
int length(path p);
This is the number of (linear or cubic) segments in path p
.
If p
is cyclic, this is the same as the number of nodes in p
.
int size(path p);
This is the number of nodes in the path p
.
If p
is cyclic, this is the same as length(p)
.
bool cyclic(path p);
returns true
iff path p
is cyclic.
bool straight(path p, int i);
returns true
iff the segment of path p
between node
i
and node i+1
is straight.
bool piecewisestraight(path p)
returns true
iff the path p
is piecewise straight.
pair point(path p, int t);
If p
is cyclic, return the coordinates of node t
mod
length(p)
. Otherwise, return the coordinates of node t
,
unless t
< 0 (in which case point(0)
is returned) or
t
> length(p)
(in which case point(length(p))
is returned).
pair point(path p, real t);
This returns the coordinates of the point between node floor(t)
and floor(t)+1
corresponding to the cubic spline parameter
t-floor(t)
(see Bezier curves). If t
lies outside the range
[0,length(p)
], it is first reduced modulo length(p)
in the case where p
is cyclic or else converted to the corresponding
endpoint of p
.
pair dir(path p, int t, int sign=0, bool normalize=true);
If sign < 0
, return the direction (as a pair) of the incoming tangent
to path p
at node t
; if sign > 0
, return the
direction of the outgoing tangent. If sign=0
, the mean of these
two directions is returned.
pair dir(path p, real t, bool normalize=true);
returns the direction of the tangent to path p
at the point
between node floor(t)
and floor(t)+1
corresponding to the
cubic spline parameter t-floor(t)
(see Bezier curves).
pair dir(path p)
returns dir(p,length(p)).
pair dir(path p, path q)
returns unit(dir(p)+dir(q)).
pair accel(path p, int t, int sign=0);
If sign < 0
, return the acceleration of the incoming path
p
at node t
; if sign > 0
, return the
acceleration of the outgoing path. If sign=0
, the mean of these
two accelerations is returned.
pair accel(path p, real t);
returns the acceleration of the path p
at the point t
.
real radius(path p, real t);
returns the radius of curvature of the path p
at the point t
.
pair precontrol(path p, int t);
returns the precontrol point of p
at node t
.
pair precontrol(path p, real t);
returns the effective precontrol point of p
at parameter t
.
pair postcontrol(path p, int t);
returns the postcontrol point of p
at node t
.
pair postcontrol(path p, real t);
returns the effective postcontrol point of p
at parameter t
.
real arclength(path p);
returns the length (in user coordinates) of the piecewise linear
or cubic curve that path p
represents.
real arctime(path p, real L);
returns the path "time", a real number between 0 and the length of
the path in the sense of point(path p, real t)
, at which the
cumulative arclength (measured from the beginning of the path) equals L
.
real arcpoint(path p, real L);
returns point(p,arctime(p,L))
.
real dirtime(path p, pair z);
returns the first "time", a real number between 0 and the length of
the path in the sense of point(path, real)
, at which the tangent
to the path has the direction of pair z
, or -1 if this never happens.
real reltime(path p, real l);
returns the time on path p
at the relative fraction l
of
its arclength.
pair relpoint(path p, real l);
returns the point on path p
at the relative fraction l
of its
arclength.
pair midpoint(path p);
returns the point on path p
at half of its arclength.
path reverse(path p);
returns a path running backwards along p
.
path subpath(path p, int a, int b);
returns the subpath of p
running from node a
to node b
.
If a
< b
, the direction of the subpath is reversed.
path subpath(path p, real a, real b);
returns the subpath of p
running from path time a
to path
time b
, in the sense of point(path, real)
. If a
<
b
, the direction of the subpath is reversed.
real[] intersect(path p, path q, real fuzz=-1);
If p
and q
have at least one intersection point, return a
real array of length 2 containing the times representing the respective
path times along p
and q
, in the sense of
point(path, real)
, for one such intersection point (as chosen by
the algorithm described on page 137 of The MetaFontbook
).
The computations are performed to the absolute error specified by fuzz
,
or if fuzz < 0
, to machine precision. If the paths do not
intersect, return a real array of length 0.
real[][] intersections(path p, path q, real fuzz=-1);
Return all (unless there are infinitely many) intersection times of
paths p
and q
as a sorted array of real arrays of length 2
(see sort). The computations are performed to the absolute error
specified by fuzz
, or if fuzz < 0
, to machine precision.
real[] intersections(path p, explicit pair a, explicit pair b, real fuzz=-1);
Return all (unless there are infinitely many) intersection times of path
p
with the (infinite) line through points a
and b
as a sorted array. The intersections returned are guaranteed to be
correct to within the absolute error specified by fuzz
, or if
fuzz < 0
, to machine precision.
real[] times(path p, real x)
returns all intersection times of path p
with the vertical line
through (x,0)
.
real[] times(path p, explicit pair z)
returns all intersection times of path p
with the horizontal line
through (0,z.y)
.
real[] mintimes(path p)
returns an array of length 2 containing times at which path p
reaches its minimal horizontal and vertical extents, respectively.
real[] maxtimes(path p)
returns an array of length 2 containing times at which path p
reaches its maximal horizontal and vertical extents, respectively.
pair intersectionpoint(path p, path q, real fuzz=-1);
returns the intersection point point(p,intersect(p,q,fuzz)[0])
.
pair[] intersectionpoints(path p, path q, real fuzz=-1);
returns an array containing all intersection points of the paths
p
and q
.
pair extension(pair P, pair Q, pair p, pair q);
returns the intersection point of the extensions of the line segments
P--Q
and p--q
, or if the lines are parallel,
(infinity,infinity)
.
slice cut(path p, path knife, int n);
returns the portions of path p
before and after the n
th
intersection of p
with path knife
as a structure
slice
(if no intersection exist is found, the entire path is
considered to be ‘before’ the intersection):
struct slice { path before,after; }
The argument n
is treated as modulo the number of intersections.
slice firstcut(path p, path knife);
equivalent to cut(p,knife,0);
Note that firstcut.after
plays the role of the MetaPost
cutbefore
command.
slice lastcut(path p, path knife);
equivalent to cut(p,knife,-1);
Note that lastcut.before
plays the role of the
MetaPost cutafter
command.
path buildcycle(... path[] p);
This returns the path surrounding a region bounded by a list of two or more
consecutively intersecting paths, following the behaviour of the
MetaPost buildcycle
command.
pair min(path p);
returns the pair (left,bottom) for the path bounding box of path p
.
pair max(path p);
returns the pair (right,top) for the path bounding box of path p
.
int windingnumber(path p, pair z);
returns the winding number of the cyclic path p
relative to the point
z
. The winding number is positive if the path encircles z
in the
counterclockwise direction. If z
lies on p
the constant
undefined
(defined to be the largest odd integer) is returned.
bool interior(int windingnumber, pen fillrule)
returns true if windingnumber
corresponds to an interior point
according to fillrule
.
bool inside(path p, pair z, pen fillrule=currentpen);
returns true
iff the point z
lies inside or on the edge of
the region bounded by the cyclic path p
according to the fill
rule fillrule
(see fillrule).
int inside(path p, path q, pen fillrule=currentpen);
returns 1
if the cyclic path p
strictly contains q
according to the fill rule fillrule
(see fillrule), -1
if the cyclic path q
strictly contains p
, and 0
otherwise.
pair inside(path p, pen fillrule=currentpen);
returns an arbitrary point strictly inside a cyclic path p
according to the fill rule fillrule
(see fillrule).
path[] strokepath(path g, pen p=currentpen);
returns the path array that PostScript
would fill in drawing path
g
with pen p
.
guide
an unresolved cubic spline (list of cubic-spline nodes and control points).
The implicit initializer for a guide is nullpath
; this is useful
for building up a guide within a loop.
A guide is similar to a path except that the computation of the cubic spline is deferred until drawing time (when it is resolved into a path); this allows two guides with free endpoint conditions to be joined together smoothly. The solid curve in the following example is built up incrementally as a guide, but only resolved at drawing time; the dashed curve is incrementally resolved at each iteration, before the entire set of nodes (shown in red) is known:
size(200); real mexican(real x) {return (1-8x^2)*exp(-(4x^2));} int n=30; real a=1.5; real width=2a/n; guide hat; path solved; for(int i=0; i < n; ++i) { real t=-a+i*width; pair z=(t,mexican(t)); hat=hat..z; solved=solved..z; } draw(hat); dot(hat,red); draw(solved,dashed);
We point out an efficiency distinction in the use of guides and paths:
guide g; for(int i=0; i < 10; ++i) g=g--(i,i); path p=g;
runs in linear time, whereas
path p; for(int i=0; i < 10; ++i) p=p--(i,i);
runs in quadratic time, as the entire path up to that point is copied at each step of the iteration.
The following routines can be used to examine the individual elements of a guide without actually resolving the guide to a fixed path (except for internal cycles, which are resolved):
int size(guide g);
Analogous to size(path p)
.
int length(guide g);
Analogous to length(path p)
.
bool cyclic(path p);
Analogous to cyclic(path p)
.
pair point(guide g, int t);
Analogous to point(path p, int t)
.
guide reverse(guide g);
Analogous to reverse(path p)
. If g
is cyclic and
also contains a secondary cycle, it is first solved to a
path, then reversed. If g
is not cyclic but contains an internal
cycle, only the internal cycle is solved before reversal. If there are
no internal cycles, the guide is reversed but not solved to a path.
pair[] dirSpecifier(guide g, int i);
This returns a pair array of length 2 containing the outgoing (in
element 0) and incoming (in element 1) direction specifiers (or
(0,0)
if none specified) for the segment of guide g
between nodes i
and i+1
.
pair[] controlSpecifier(guide g, int i);
If the segment of guide g
between nodes i
and i+1
has explicit outgoing and incoming control points, they are returned as
elements 0 and 1, respectively, of a two-element array. Otherwise, an
empty array is returned.
tensionSpecifier tensionSpecifier(guide g, int i);
This returns the tension specifier for the segment of guide g
between
nodes i
and i+1
. The individual components of the
tensionSpecifier
type can be accessed as the virtual members
in
, out
, and atLeast
.
real[] curlSpecifier(guide g);
This returns an array containing the initial curl specifier (in element 0)
and final curl specifier (in element 1) for guide g
.
As a technical detail we note that a direction specifier given to
nullpath
modifies the node on the other side: the guides
a..{up}nullpath..b; c..nullpath{up}..d; e..{up}nullpath{down}..f;
are respectively equivalent to
a..nullpath..{up}b; c{up}..nullpath..d; e{down}..nullpath..{up}f;
Next: Pens, Previous: Data types, Up: Programming [Contents][Index]