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fiziko.mp
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fiziko.mp
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% fiziko 0.2.0
% MetaPost library for physics textbook illustrations
% Copyright 2022 Sergey Slyusarev
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <https://www.gnu.org/licenses/>.
% https://github.com/jemmybutton/fiziko
%
% Here we define some things of general interest
%
pi := 3.1415926;
radian := 180/pi;
vardef sin primary x = (sind(x*radian)) enddef;
vardef cos primary x = (cosd(x*radian)) enddef;
vardef log (expr n, b) =
save rv;
numeric rv;
if n > 0:
rv := (mlog(n)/mlog(b));
else:
rv := 0;
fi;
rv
enddef;
vardef arcsind primary x = angle((1+-+x,x)) enddef;
vardef arccosd primary x = angle((x,1+-+x)) enddef;
vardef arcsin primary x = ((arcsind(x))/radian) enddef;
vardef arccos primary x = ((arccosd(x))/radian) enddef;
vardef angleRad primary x = angle(x)/radian enddef;
vardef dirRad primary x = dir(x*radian) enddef;
% used here and there.
vardef sign (expr x)=
if x > 0: 1 fi
if x < 0: -1 fi
if x = 0: 1 fi
enddef;
% This is inverted `clip`
primarydef i maskedWith p =
begingroup
save q, invertedmask, resultimage, breakpoint;
pair q[];
path invertedmask;
picture resultimage;
resultimage := i;
q1 := ulcorner(i) shifted (-1, 1);
q3 := lrcorner(i) shifted (1, -1);
q2 := (xpart(q3), ypart(q1));
q4 := (xpart(q1), ypart(q3));
breakpoint := ypart((ulcorner(p)--llcorner(p)) firstIntersectionTimes p);
invertedmask := (subpath (breakpoint, length(p) + breakpoint) of p) -- q1 -- q2 -- q3 -- q4 -- q1 -- cycle;
clip resultimage to invertedmask;
resultimage
endgroup
enddef;
%
% Since metapost is somewhat unpredictable in determining where paths intersect, here's macro
% that returns first intersection times with first path (ray) priority.
% Actually, it is so in most cases, but sometimes second path can take precedence,
% so the macro just checks whether reversing 'q' changes something
%
primarydef p firstIntersectionTimes q =
begingroup
save t;
pair t[];
t1 := p intersectiontimes q;
t2 := p intersectiontimes reverse(q);
if xpart(t1) < xpart(t2):
t3 := t1;
else:
t3 := (xpart(t2), length(q) - ypart(t2));
fi;
if xpart(t1) < 0: t3 := t2; fi;
t3
endgroup
enddef;
% This checks if point a is inside of closed path p
primarydef a isInside p =
begingroup
save ang, v, i, rv, pp;
boolean rv;
pair pp[];
ang := 0;
for i := 0 step 1/4 until (length(p)):
pp1 := (point i of p) - a;
pp2 := (point i + 1/4 of p) - a;
if (pp1 <> (0, 0)) and (pp2 <> (0, 0)):
v := angle(pp1) - angle(pp2);
if v > 180: v := v - 360; fi; if v < -180: v := v + 360; fi;
ang := ang + v;
fi;
endfor;
if abs(ang) > 355:
rv := true;
else:
rv := false;
fi;
rv
endgroup
enddef;
% rotation in radians
primarydef somethingToRotate radRotated radAngle =
somethingToRotate rotated ((radAngle/pi)*180)
enddef;
%
% some 3D stuff
%
% this one's from byrne.mp
primarydef colorone dotprodXYZ colortwo =
begingroup
save xp, yp, zp;
numeric xp[], yp[], zp[];
xp1 := (redpart colorone);
yp1 := (greenpart colorone);
zp1 := (bluepart colorone);
xp2 := (redpart colortwo);
yp2 := (greenpart colortwo);
zp2 := (bluepart colortwo);
xp1*xp2 + yp1*yp2 + zp1*zp2
endgroup
enddef;
%
% sometimes it's useful to put some arrows along the path. this macro puts them
% in the middles of the segments that have length no less than midArrowLimit;
%
midArrowLimit := 1cm;
def drawmidarrow (expr p) text t =
begingroup
save i, j, q;
path q;
j := 0;
for i := 1 upto length(p):
if arclength(subpath(i-1, i) of p) >= midArrowLimit:
q := subpath(j, i - 1/2) of p;
j := i - 1/2;
draw q t;
filldraw arrowhead q t;
fi;
endfor;
draw subpath(j, length(p)) of p t;
endgroup
enddef;
% This macro marks angles, unsurprisingly
def markAngle (expr a, o, b) (text t) =
begingroup
save p, an, d;
numeric an[], d[];
pair p;
an1 := angle(a-o);
an2 := angle(b-o) - an1;
if (an2 < 0): an2 := an2 + 360; fi;
an3 := an1 + 1/2an2;
p := center(t);
d1 := abs(ulcorner(t)-lrcorner(t));
if (an2 < 90) and (an2 > 0):
d2 := max(1/3cm, (d1/(abs(sind(an2))*1/3cm))*1/3cm);
else:
d2 := 1/3cm;
fi;
draw subpath (0, 8an2/360) of fullcircle scaled 2d2 rotated an1 shifted o withpen thinpen;
draw (t) shifted -p shifted o shifted (dir(an3)*(d2 + d1));
endgroup
enddef;
%
% Here we define some auxilary global variables
%
% Offset path algorithm can subdivide original path in order to be more precise
offsetPathSteps := 4;
% The following macro sets all the values related to minimal stroke width at once.
% It can be used to easily redefine all of them.
def defineMinStrokeWidth (expr msw) =
% We don't want to display strokes that are too thin to print. Default value
% is subject to change when needed.
minStrokeWidth := msw;
maxShadingStrokeWidth := 3/2minStrokeWidth;
% At some point it's useless to display even dashes
minDashStrokeWidth := 1/3minStrokeWidth;
% this value corresponds to particular dashing algorithm and is subject to change whenever this algorithm changes
minDashStrokeLength := 6minStrokeWidth;
dashStrokeWidthStep := (minStrokeWidth - minDashStrokeWidth)/16;
% all the shading algorithms need to know how close lines should be packed
shadingDensity := 3maxShadingStrokeWidth;
stippleSize := 3/2minStrokeWidth;
minStippleStep := 1/2stippleSize;
stippleShadingDensity := 3minStippleStep;
minStippleStrokeWidth := 1/20stippleSize;
% here are some pens
pen thinpen, thickpen, fatpen, stipplepen;
thinpen := pencircle scaled minStrokeWidth;
thickpen := pencircle scaled 3minStrokeWidth;
fatpen := pencircle scaled 6minStrokeWidth;
stipplepen := pencircle scaled stippleSize;
enddef;
defineMinStrokeWidth(1/5pt);
% here we set global light direction
def defineLightDirection (expr ldx, ldy) =
pair lightDirection;
color lightDirectionVectorXYZ;
lightDirection := (ldx, ldy);
lightDirectionVectorXYZ := (0, 0, 1);
lightDirectionVectorXYZ := rotateXYZaround.x(lightDirectionVectorXYZ, ldy);
lightDirectionVectorXYZ := rotateXYZaround.y(lightDirectionVectorXYZ, ldx);
enddef;
vardef rotateXYZaround@# (expr p, a) =
save partProj, rv;
pair partProj;
color rv;
if str @# = "x":
partProj := (greenpart(p), bluepart(p)) radRotated -a;
rv := (redpart(p), xpart(partProj), ypart(partProj));
elseif str @# = "y":
partProj := (redpart(p), bluepart(p)) radRotated -a;
rv := (xpart(partProj), greenpart(p), ypart(partProj));
elseif str @# = "z":
partProj := (redpart(p), greenpart(p)) radRotated -a;
rv := (xpart(partProj), ypart(partProj), bluepart(p));
else:
errmessage("What axis is " & str @# & "?");
fi;
rv
enddef;
defineLightDirection(-1/8pi, 1/8pi);
boolean shadowsEnabled;
shadowsEnabled := false;
%
% To simplify further calculations we need subdivided original path
%
vardef pathSubdivideBase (expr p, subdivideStep, i) =
save returnPath, sp;
path returnPath, sp;
returnPath := point i of p;
if i<length(p):
sp := subpath(i, i + subdivideStep) of p;
returnPath := returnPath .. controls (postcontrol 0 of sp) and (precontrol 1 of sp) .. pathSubdivideBase (p, subdivideStep, i + subdivideStep);
fi;
if (i = 0) and (cycle p):
(subpath(0, length(returnPath)-1) of returnPath) .. controls (postcontrol length(returnPath)-1 of returnPath) and (precontrol length(returnPath) of returnPath) .. cycle
else:
returnPath
fi
enddef;
vardef offsetPathSubdivide (expr p) =
pathSubdivideBase(p, 1/offsetPathSteps, 0)
enddef;
vardef pathSubdivide (expr p, n) =
pathSubdivideBase(p, 1/n, 0)
enddef;
%
% This macro creates a template offset path to a straight line, so we can correct angles
% It might appear as if we need to calculate derivative of the function somehow, instead of mocking it
% but this function might be anything, function of coordinates of distance to some point etc.,
% so consider this a lazy way to do the right thing.
%
% either offsetPathTime or offsetPathLength are intended to be used as arguments. offsetPathTime is for time and offsetPathLength is for distance
%
vardef offsetPathTemplate (expr p, i) (text offsetFunction) =
save returnPath, offsetPathTime, offsetPathLength, instantDirection, nextDirection;
numeric offsetPathTime, offsetPathLength, currentAngle;
pair instantDirection, nextDirection;
path returnPath;
if (i <= length(p)):
offsetPathTime := i;
else:
offsetPathTime := length(p);
fi;
if (arclength(p) > 0):
offsetPathLength := arclength(subpath (0, i) of p)/arclength(p);
else:
offsetPathLength := 0;
fi;
returnPath := (arclength(subpath (0, i) of p), offsetFunction);
if (i < length(p)):
% this thing is glitchy, but should be more accurate
%if (arclength(subpath (0, i) of p) < arclength(subpath (0, i + 1/4) of p)):
% offsetPathTime := i + 1/4;
% offsetPathLength := arclength(subpath (0, i + 1/4) of p)/arclength(p);
% instantDirection := unitvector((arclength(subpath (0, i + 1/4) of p), offsetFunction) - point 0 of returnPath);
% offsetPathTime := i + 1;
% offsetPathLength := arclength(subpath (0, i + 1) of p)/arclength(p);
% nextDirection := (arclength(subpath (0, i + 1) of p), offsetFunction);
% offsetPathTime := i + 3/4;
% offsetPathLength := arclength(subpath (0, i + 3/4) of p)/arclength(p);
% nextDirection := unitvector(nextDirection - (arclength(subpath (0, i + 3/4) of p), offsetFunction));
% returnPath := returnPath{instantDirection} .. {nextDirection}offsetPathTemplate(p, i + 1)(offsetFunction);
% returnPath := returnPath -- offsetPathTemplate(p, i + 1)(offsetFunction);
%else:
returnPath := returnPath -- offsetPathTemplate(p, i + 1)(offsetFunction);
%fi;
fi;
returnPath
enddef;
%
% This macro creates offset path p based on previously built template q, instead of function itself
% It is loosely based on something called Tiller-Hanson heuristic as described here:
% https://math.stackexchange.com/questions/465782/control-points-of-offset-bezier-curve
%
vardef offsetPathGenerate (expr p, q, i) =
save returnPath, c, d, pl, ps;
path returnPath, pl[];
pair c[], d[];
c1 := precontrol i of p;
c2 := point i of p;
c3 := postcontrol i of p;
if abs(c1-c2) = 0:
c1 := c2 shifted (c2-c3);
fi;
if abs(c3-c2) = 0:
c3 := c2 shifted (c2-c1);
fi;
if (abs(c1-c2) > 0) and (abs(c2-c3) > 0):
d1 := unitvector(c1-c2) rotated -90;
d2 := unitvector(c2-c3) rotated -90;
pl1 := (unitvector(c2-c1)--unitvector(c1-c2))
scaled arclength(subpath (i - 1/2, i + 1/2) of p)
shifted (point i of p shifted (d1 scaled ypart(point i of q)));
pl2 := (unitvector(c2-c3)--unitvector(c3-c2))
scaled arclength(subpath (i - 1/2, i + 1/2) of p)
shifted (point i of p shifted (d2 scaled ypart(point i of q)));
if (abs(angle(d1) - angle(d2)) > 2) and (xpart(pl1 intersectiontimes pl2) > 0):
c4 := pl1 intersectionpoint pl2;
else:
c4 := c2 shifted (d1 scaled ypart(point i of q));
fi;
returnPath := c4;
else:
returnPath := c2 shifted (unitvector( (point i-1 of p) - (point i+1 of p) rotated -90) scaled ypart (point i of q));
fi;
if i < length(p):
path ps;
ps := subpath (i, i + 1) of p;
c1 := point 0 of ps;
c2 := postcontrol 0 of ps;
c3 := precontrol 1 of ps;
c4 := point 1 of ps;
c5 := point 0 of returnPath;
if (abs(c3-c4)>0)
and (abs(c1-c2)>0)
and (abs(c1-c4)>0)
and (abs(direction i of q) > 0):
c6 := c4 shifted (unitvector(c4 - c3) rotated 90 scaled ypart(point i + 1 of q));
c7 := (c2 - c1) scaled (abs(c5-c6)/abs(c1-c4)) rotated angle(direction i of q) shifted c5;
c8 := (c3 - c4) scaled (abs(c5-c6)/abs(c1-c4)) rotated angle(direction i + 1 of q) shifted c6;
returnPath := returnPath .. controls c7 and c8 .. offsetPathGenerate (p, q, i + 1);
else:
returnPath := returnPath -- offsetPathGenerate (p, q, i + 1);
fi;
fi;
returnPath
enddef;
%
% Frontend for offsetPathGenerate and offsetPathTemplate
%
vardef offsetPath (expr p)(text offsetFunction) =
offsetPathGenerate (p, offsetPathTemplate(p, 0)(offsetFunction), 0)
enddef;
%
% Brush macro. It draws line with brush of variable width.
% For parts thicker than minStrokeWidth it uses offsetPath functions'
% results, for thiner parts it draws dashed lines of fixed width
%
def brushGenerate (expr p, q, i) =
begingroup
save w, brushPath, bt, t;
numeric w[], t[];
path brushPath[], bt;
bt := q;
w0 := (ypart(urcorner(bt)));
w1 := (ypart(lrcorner(bt)));
t := cutPathTime(bt, minStrokeWidth);
if ((w0 > minStrokeWidth)
and (w1 < minStrokeWidth)
and (t > 0)
and (t < length(p))
and (arclength(p) > minDashStrokeLength)
and (i < 10)):
brushGenerate (subpath (0, t) of p, subpath (0, t) of q, i + 1);
brushGenerate (subpath (t, length(p)) of p, subpath (t, length(q)) of q, i + 1);
elseif (arclength(p) > 0):
if (w0 > 99/100minStrokeWidth)
and (w1 > 99/100minStrokeWidth):
brushPath1 := offsetPathGenerate (p, q yscaled 1/2, 0);
brushPath2 := offsetPathGenerate (p, q yscaled -1/2, 0);
fill brushPath1 -- reverse(brushPath2) -- cycle;
elseif (w0 < 101/100minStrokeWidth) and (w1 < 101/100minStrokeWidth):
thinBrushGenerate (p, q, 0)
fi;
fi;
endgroup
enddef;
%
% macro for thin lines which are actually dashed
%
vardef thinBrushGenerate@#(expr p, q, i) =
begingroup
save w, brushPath, bt, t, h, minLength, minWidth, dashPatternImage;
numeric w[], t[];
path brushPath[], bt;
picture dashPatternImage;
if (str @# = "") or (str @# = "hatches"):
minLength := minDashStrokeLength;
minWidth := minDashStrokeWidth;
elseif str @# = "stipples":
minLength := minStippleStep;
minWidth := minStippleStrokeWidth;
fi;
bt := q;
w0 := ypart(urcorner(bt));
w1 := ypart(lrcorner(bt));
if (w0 > minWidth + 1/100):
if (w1 > minWidth - 1/100):
w2 := floor((1/2(w0 + w1) - minWidth)/dashStrokeWidthStep)*dashStrokeWidthStep + minWidth;
else:
w2 := minWidth;
fi;
t := cutPathTime(bt, w2);
brushPath1 := subpath (0, t) of p;
brushPath2 := subpath (t, length(p)) of p;
if (((w0 - w1) >= dashStrokeWidthStep) and (i < 15))
and ((arclength(brushPath1) > minLength)
or (arclength(brushPath2) > minLength))
and (t > 1/100) and (t < length(p) - 1/100):
thinBrushGenerate@#(brushPath1, subpath (0, t) of q, i + 1);
thinBrushGenerate@#(brushPath2, subpath (t, length(q)) of q, i + 1);
else:
if (str @# = "") or (str @# = "hatches"):
if (w2 > minStrokeWidth):
w2 := minStrokeWidth;
fi;
if (w2 >= minWidth) and (arclength(p) > 0):
if (w2 < minStrokeWidth) and (arclength(p) > minLength):
draw p withpen thinpen dashed thinBrushPattern(w2, arclength(p));
else:
draw p withpen thinpen;
fi;
fi;
elseif str @# = "stipples":
begingroup
interim linecap := rounded;
save stippleSizeVar;
stippleSizeVar := stippleSize;
save stippleSize;
w2 := 1/3w2;
if (w2 >= minWidth) and (arclength(p) > 0):
stippleSize := stippleSizeVar * (0.9 + uniformdeviate(0.3));
dashPatternImage := stipplesBrushPattern(w2, arclength(p));
if urcorner(dashPatternImage) <> (0,0):
brushPath1 := offsetPathGenerate (p, (q yscaled 0) shifted (0, 1/3stippleShadingDensity), 0);
draw brushPath1 withpen (pencircle scaled stippleSize) dashed dashPatternImage;
fi;
stippleSize := stippleSizeVar * (0.9 + uniformdeviate(0.3));
dashPatternImage := stipplesBrushPattern(w2, arclength(p));
if urcorner(dashPatternImage) <> (0,0):
brushPath2 := offsetPathGenerate (p, (q yscaled 0) shifted (0, -1/3stippleShadingDensity), 0);
draw brushPath2 withpen (pencircle scaled stippleSize) dashed dashPatternImage;
fi;
stippleSize := stippleSizeVar * (0.9 + uniformdeviate(0.3));
dashPatternImage := stipplesBrushPattern(w2, arclength(p));
if urcorner(dashPatternImage) <> (0,0):
draw p withpen (pencircle scaled stippleSize) dashed dashPatternImage;
fi;
fi;
endgroup
fi;
fi;
fi;
endgroup
enddef;
%
% this macro returns path as a shaded edge
%
vardef shadedEdge (expr p) =
image(
brushGenerate (p,
offsetPathTemplate (p, 0) (
1/2minStrokeWidth + 2*minStrokeWidth
* normalVectorToLightness(
sphereAnglesToNormalVector(
(angleRad(point offsetPathTime of p), arcsin(1/2))
), 0, point offsetPathTime of p
)
), 0);
)
enddef;
%
% Whenever we have brush thinner than minStrokeWidth we call this dash pattern macro
%
vardef thinBrushPattern (expr w, l) =
save d;
numeric d[];
d0 := w;
if d0 > minStrokeWidth: d0 := minStrokeWidth; fi;
% d1 is a result of some arbitrary function of line width
% we do not use simple linear function because minimal dash length
% also shouldn't be less than minStrokeWidth.
% After we get d1 other measurements are calculated,
% so filled area per unit length remains adequate and dashes are aligned
% with segments
% d1*mSW = (d1*w + d2*w) -> d2=d1(mSW-w)/w
d1 := (1/2minDashStrokeLength) + (((d0/minStrokeWidth)**(5/2))*1/2minDashStrokeLength);
d1 := d1 + 1/2uniformdeviate(d1);
d2 := (minStrokeWidth - d0)*(d1/d0);
d3 := round(l/(d2 + d1));
if (d3 < 1): d3 := 1; fi;
d4 := (l/d3)/(d2 + d1);
d1 := d1*d4;
d2 := d2*d4;
if (uniformdeviate(2) > 1):
dashpattern (on d1/2 off d2 on d1/2)
else:
dashpattern (off d2/2 on d1 off d2/2)
fi
enddef;
%
% Stipples are also dashes
%
vardef stipplesBrushPattern (expr w, l) =
save d, n, rn, rv, ss;
ss := 1/1000;
numeric d[];
picture rv;
%if w > stippleSize:
% d0 := minStippleStep;
%else:
n := (w*l)/(stippleSize**2);
rn := floor(n);
if rn > 0:
d0 := l/rn;
fi;
%fi;
if rn > 0:
d1 := uniformdeviate(d0);
d2 := d0-d1;
if rn >=3:
d3 := uniformdeviate(d0);
d4 := d0-d3;
%if uniformdeviate(2) > 1:
% d5 := uniformdeviate(d0);
% d6 := d0-d5;
% rv := dashpattern (off d1 on ss off (d2+d5)-ss on ss off (d4+d6)-ss on ss off d3-ss);
%else:
rv := dashpattern (off d1 on ss off (d2+d3)-ss on ss off d4-ss);
%fi;
else:
rv := dashpattern (off d1 on ss off d2-ss);
fi;
else:
if uniformdeviate(1) < n:
rv := dashpattern (off uniformdeviate(l-ss) on ss off l);
else:
rv := image();
fi;
fi;
rv
enddef;
%
% macro that actually draws line of variable width
%
vardef brush (expr p) (text offsetFunction) =
image(
brushGenerate (p, offsetPathTemplate(p, 0)(offsetFunction), 0);
)
enddef;
%
% same, but only for thin brushes
%
vardef thinBrush@#(expr p) (text offsetFunction) =
image(
thinBrushGenerate@#(p, offsetPathTemplate(p, 0)(offsetFunction), 0);
)
enddef;
%
% This macro generates tube between paths p and q, of variable width d
% Tube is subdivided into segments in such a way that within every segment
% we need 2**n lines to generate even fill
%
vardef tubeGenerate@#(expr p, q, d, i) =
save w, bw, k, t, tubeWidth, sp, currentPath, currentTubePath, currentDepth, lineDensity;
numeric w[], bw[], t, currentDepth;
path tubeWidth, sp, currentPath, currentTubePath;
tubeWidth := d yscaled 2;
if (str @# = "") or (str @# = "hatches"):
lineDensity := shadingDensity;
elseif (str @# = "stipples"):
lineDensity := stippleShadingDensity;
fi;
w0 := (ypart(urcorner(tubeWidth))) - 1/1000;
w1 := (ypart(lrcorner(tubeWidth))) + 1/1000;
w2 := ceiling(log(w0/lineDensity, 2));
w3 := ceiling(log(w1/lineDensity, 2));
if ((w2 > w3) and (i<20)):
t := cutPathTime(tubeWidth, lineDensity*(2**(w2-1)));
tubeGenerate@#(subpath (0, t) of p, subpath (0, t) of q, subpath (0, t) of d, i + 1);
tubeGenerate@#(subpath (t, length(p)) of p, subpath (t, length(q)) of q, subpath (t, length(d)) of d, i + 1);
else:
if (arclength(p) > 0) and (arclength(q) > 0):
bw1 := 2**w2;
currentTubePath := interpath (1/2, q, p);
for k := 0 upto bw1:
currentPath := interpath (k/bw1, q, p);
angleOnTube := arcsin(((k/bw1)*2) - 1);
currentDepth := -abs((1-sin(angleOnTube + 1/2pi))*w0);
if shadowsEnabled:
currentPath := shadowCut(currentPath, currentDepth);
fi;
if (str @# = "") or (str @# = "hatches"):
brushGenerate (currentPath,
offsetPathTemplate(currentPath, 0)(
maxShadingStrokeWidth
if odd (k): * (abs(ypart(point offsetPathTime of tubeWidth)/bw1) - 1/2lineDensity) fi
* normalVectorToLightness(
tubeAnglesToNormalVector((
angleOnTube,
angleRad(direction offsetPathTime of currentTubePath),
angleRad(direction offsetPathTime of (tubeWidth yscaled 1/2))
)), currentDepth, point offsetPathTime of currentPath)
), 0);
elseif (str @# = "stipples"):
begingroup
save stippleShadingDensity;
if w2 > 0:
stippleShadingDensity := 2w0/(2**w2); % When the distance between the lines changes, wtipples should spread further apart
else:
stippleShadingDensity := w0;
fi;
thinBrushGenerate.@#(currentPath,
offsetPathTemplate(currentPath, 0)(
stippleSize
if odd (k): * (abs(ypart(point offsetPathTime of tubeWidth)/bw1) - 1/2lineDensity) fi
* normalVectorToLightness(
tubeAnglesToNormalVector((
angleOnTube,
angleRad(direction offsetPathTime of currentTubePath),
angleRad(direction offsetPathTime of (tubeWidth yscaled 1/2))
)), currentDepth, point offsetPathTime of currentPath)
), 0);
endgroup
fi;
endfor;
fi;
fi;
enddef;
%
% This macro is analogous to tubeGenerate, but draws transverse strokes
% result is somewhat suboptimal for now, but in simple cases it works ok
%
def tubeGenerateAlt (expr p, q, d) =
begingroup
save spth, lpth, currentPath, pos, t, pthdir, corr, o, l, i, j, k, tubeAngle, pathAngle, scorr, dt;
numeric l[];
path spth, lpth, currentPath;
pos := 0;
j := 0;
forever:
dt := (xpart(point pos of d) + 1/2shadingDensity);
scorr := cosd(angle(direction xpart(d intersectiontimes ((dt, ypart(lrcorner(d))) -- (dt, ypart(urcorner(d))))) of d));
t1 := arctime ((arclength(subpath(0, pos) of p)) + shadingDensity/scorr) of p;
t2 := arctime ((arclength(subpath(0, pos) of q)) + shadingDensity/scorr) of q;
if (arclength(subpath(pos, t1) of p) < arclength(subpath(pos, t1) of q)):
pthdir := -1;
t3 := t1;
else:
pthdir := 1;
t3 := t2;
fi;
corr := round(arclength(subpath(pos, t3) of if pthdir = 1: p else: q fi)/(shadingDensity/scorr));
if (corr < 1): corr := 1; fi;
corr := (arclength(subpath(pos, t3) of if pthdir = 1: p else: q fi) - (corr*(shadingDensity/scorr)))/corr;
t3 := arctime (arclength(subpath(0, t3) of if pthdir = 1: q else: p fi) - 1/3corr) of if pthdir = 1: q else: p fi;
spth := subpath(pos, t3) of if pthdir = 1: q else: p fi;
lpth := subpath(pos, t3) of if pthdir = 1: p else: q fi;
tubeAngle := angleRad(direction 1/2[pos, t3] of d);
pathAngle := angleRad(direction 1/2 of interpath (1/2, spth, lpth));
pos := t3;
l1 := round(arclength(lpth)/(shadingDensity/scorr));
if (l1 < 1): l1 := 1; fi;
l2 := arclength(lpth)/(l1*(shadingDensity/scorr));
for i := 0 upto l1 - 1:
j := j + 1;
k := i*(arclength(lpth)/l1);
currentPath := point (arctime k of lpth) of spth -- point (arctime k of lpth) of lpth;
currentPath := offsetPathSubdivide(currentPath);
brushGenerate (
currentPath,
offsetPathTemplate(currentPath, 0)(
maxShadingStrokeWidth
* orderFade(offsetPathLength[1/l1, l2], j)
* normalVectorToLightness(
tubeAnglesToNormalVector((
arcsin(pthdir*((offsetPathLength*2)-1)),
pathAngle,
tubeAngle)
), -2(1/2arclength(currentPath))+sqrt(1 - (2offsetPathLength - 1)**2)*(1/2arclength(currentPath)), point offsetPathTime of currentPath)
)
, 0);
endfor;
exitif pos >= length(p);
endfor;
endgroup
enddef;
%
% This macro converts some measurements of point on tube to absolute angle.
% Since there are three such measurements, macro gets them as as a single
% argument of "color" type, in case it would eventually appear as a result
% of some other macro.
%
% redpart is the angle on the tube's circumference
% greenpart is the angle of the tube path
% bluepart is the angle of the tube's outline
%
vardef tubeAnglesToNormalVector (expr p) =
save normalVector;
color normalVector;
normalVector := (0, 0, 1);
normalVector := rotateXYZaround.y(normalVector, -bluepart(p));
normalVector := rotateXYZaround.x(normalVector, redpart(p));
normalVector := rotateXYZaround.z(normalVector, -greenpart(p));
normalVector
enddef;
%
% frontend to simplify tube drawing. tubeOutline variable changes on every call
% of the function and can be used afterwards.
%
path tubeOutline;
boolean drawTubeEnds;
drawTubeEnds := true;
vardef tube@#(expr p)(text offsetFunction)=
save q, respic;
path q[];
picture respic;
q0 := offsetPathSubdivide(p);
q1 := offsetPathTemplate(q0, 0)(offsetFunction);
q2 := offsetPathGenerate (q0, q1, 0);
q3 := offsetPathGenerate (q0, q1 yscaled -1, 0);
tubeOutline := q3--reverse(q2)--cycle;
if str @# = "e":
if not drawTubeEnds:
image(
draw q2 withpen thinpen;
draw q3 withpen thinpen;
)
else:
tubeOutline
fi
else:
image(
if str @# = "l":
respic := image(tubeGenerate (q2, q3, q1, 0););
elseif str @# = "s":
respic := image(tubeGenerate.stipples(q2, q3, q1, 0);)
elseif str @# = "t":
respic := image(tubeGenerateAlt (q2, q3, q1););
fi;
if (cycle p) or (not drawTubeEnds):
draw q2 withpen thinpen;
draw q3 withpen thinpen;
else:
draw q2--reverse(q3)--cycle withpen thinpen;
fi;
clip respic to (q2--reverse(q3)--cycle);
draw respic;
)
fi
enddef;
%
% Sphere can be used as a cap for a tube, so it has same 2**n lines.
%
vardef sphere@#(expr d) =
save currentCircle, origCircle, currentRadius, currentDepth, order, circleThickness, lineDensity, shadingPicture;
path currentCircle, origCircle;
numeric currentRadius, currentDepth, order;
picture shadingPicture;
if (str @# = "") or (str @# = "c"):
lineDensity := shadingDensity;
elseif (str @# = "s"):
lineDensity := stippleShadingDensity;
fi;
origCircle := fullcircle;
order := 2**ceiling(log((1/2d)/lineDensity, 2));
image(
draw fullcircle scaled d withpen thinpen;
shadingPicture := image(
for i := 1 upto order:
currentRadius := i/order;
currentCircle := origCircle scaled (currentRadius*d) rotated uniformdeviate (1/4pi);
if odd(i):
circleThickness := maxShadingStrokeWidth * ((abs(d - (lineDensity*order)))/order);
else:
circleThickness := maxShadingStrokeWidth;
fi;
currentDepth:= -(1-sqrt(1-currentRadius**2))*(1/2d);
if shadowsEnabled:
currentCircle := shadowCut(currentCircle, currentDepth);
fi;
if (str @# = "") or (str @# = "c"):
brushGenerate (currentCircle,
offsetPathTemplate (currentCircle, 0) (
circleThickness
* normalVectorToLightness(
sphereAnglesToNormalVector(
(angleRad(point offsetPathTime of currentCircle), arcsin(currentRadius))
), currentDepth, point offsetPathTime of currentCircle
)
), 0);
elseif (str @# = "s"):
begingroup
save stippleShadingDensity;
if order > 0:
stippleShadingDensity := d/order; % When the distance between the lines changes, wtipples should spread further ap
else:
stippleShadingDensity := 1/2d;
fi;
thinBrushGenerate.stipples(currentCircle,
offsetPathTemplate (currentCircle, 0) (
circleThickness
* normalVectorToLightness(
sphereAnglesToNormalVector(
(angleRad(point offsetPathTime of currentCircle), arcsin(currentRadius))
), currentDepth, point offsetPathTime of currentCircle
)
), 0);
endgroup
fi;
endfor;
);
clip shadingPicture to (fullcircle scaled d);
draw shadingPicture;
)
enddef;
%
% Alternative sphere macro. It's all about latitudinal strokes.
% The idea is: when we have a sphere with evenly distributed parallel strokes
% we know how their density rises towards edge in a projection,
% so all we need to do is to fade lines correspondingly
%
vardef sphereLat (expr d, lat) =
save p, a, x, y, sphlat, latrad, n, c, currentPath, nline, tlat;
path p[], currentPath, currentArc;
sphlat := 0;
nline := 0;
latrad := (2pi*lat/360);
n := ceiling((pi*1/2d)/shadingDensity);
if (cosd(lat) <> 0): tlat := (sind(lat)/cosd(lat)); fi;
image(
draw fullcircle scaled d withpen thinpen;
p0 := fullcircle rotated 90;
for nline := 1 upto n-1:
sphlat := nline*(pi/n);
if (sphlat + latrad < pi) and (sphlat + latrad > 0):
if (cosd(lat) <> 0):
if (sin(sphlat) <> 0):
x := tlat*(cos(sphlat)/sin(sphlat));
else:
x := 0;
fi;
else:
if ((sphlat > 1/2pi) and (lat > 0)) or ((sphlat < 1/2pi) and (lat < 0)):
x := -2;
else:
x := 2;
fi;
fi;
if (abs(x) <= 1):
y := arcsin(x);
p1 := subpath(6 + 8y/2pi, 2 - 8y/2pi) of p0;
else:
p1 := p0;
fi;
if (x > -1) and (arclength(p1) > 0):
currentPath := (p1 scaled (d*sin(sphlat)) yscaled sind(lat)) shifted (0, 1/2d*cos(sphlat)*cosd(lat));
currentPath := offsetPathSubdivide(currentPath);
brushGenerate(currentPath,
offsetPathTemplate(currentPath, 0)(
maxShadingStrokeWidth * orderFade(
sqrt(1 -
abs(
ypart(point offsetPathTime of currentPath)/(1/2d),
1 - abs(
1 - abs(
xpart(point offsetPathTime of currentPath)
/(1/2d)
)
)**abs(sind(lat))
)**2)
, nline)
* normalVectorToLightness(
sphereAnglesToNormalVector((
(
if (abs(point offsetPathTime of currentPath) > 0):
angleRad(point offsetPathTime of currentPath)
else:
0
fi
), arcsin(2abs(point offsetPathTime of currentPath)/(d+1)))
), 0, point offsetPathTime of currentPath)
), 0);
fi;
fi;
endfor;
)
enddef;
vardef orderFade (expr v, n)=
save o;
if (v > 1/256):
o := 2**ceiling(log(1/v, 2));
if ((n mod 1/2o) = 0):
if ((n mod o) = 0):