# [metapost] Re: Honza's puzzler

L. Nobre G. nobre at lince.cii.fc.ul.pt
Tue Jan 18 22:35:23 CET 2005

```Greetings,

I'm arriving at the MetaPost mailing list, for the following reason:
--------------------------------------------------------------------
On Tue, 18 Jan 2005, Larry Siebenmann wrote:

[...]
> (c) Post your programs on MP list
>
> !!
--------------------------------------------------------------------

Larry was commenting on a Metapost program that I wrote to answer a
message posted by:
Honza Prachar pracj3am at mbox.troja.mff.cuni.cz
Mon Jan 17 14:06:11 CET 2005
http://tug.org/pipermail/metapost/2005-January/000120.html
and challenged by:
Laurence Finston lfinsto1 at gwdg.de
Mon Jan 17 18:36:51 CET 2005
http://tug.org/pipermail/metapost/2005-January/000123.html
He said:
> However, I have every faith that one of the good mathematicians on this
> list can give you some hints, and maybe even a canned solution.
>
> Laurence

I'm no mathematician but I've got a canned solution:
--------------------------------------------------------------------
% plaintangency.mp
% L. Nobre G.
% IYP (2005)

input mp-tool;

def paircrossprod(expr A, B) =
( (xpart A)*(ypart B) - (xpart B)*(ypart A) )
enddef;

def firsttangencypoint( expr Path, Point, ResolvN ) =
begingroup
save auxp, i, cutp, va, vb;
path auxp;
numeric i;
pair cutp, va, vb;
auxp =
hide( va := unitvector( point 0 of Path - Point );
vb := unitvector( direction 0 of Path ); )
( paircrossprod( va, vb ), 0 )
for i=1/ResolvN step 1/ResolvN until length Path:
hide( va := unitvector( point i of Path - Point );
vb := unitvector( direction i of Path ); )
...( paircrossprod( va, vb ), i )
endfor;
cutp = auxp intersectionpoint ( origin--( 0, length Path ) );
( point ( ypart cutp ) of Path )
endgroup
enddef;

beginfig(0);
numeric u, i;
u = 5mm;
pen a, b, c;
a = pencircle scaled 3pt;
b = pencircle scaled 5pt;
c = pencircle scaled 1pt;
z1 = (1u,1u);
z2 = (4u,4u);
z3 = (4u,5u);
z4 = (3u,5u);
z5 = (3u,6u);
z6 = (4u,7u);
z7 = (6u,1u);
path cp;
cp = z1{up}..z2..z3..z4..z5..{up}z6;
draw cp withpen c;
for i=1 upto 6:
draw z[i] withpen a withcolor 0.5*(red+green);
endfor;
z8 = firsttangencypoint( cp, z7, 5 );
draw z7 withpen b withcolor green;
draw z7--z8 withpen c withcolor blue;
endfig;

beginfig(1);
numeric u;
u = 5mm;
pen a, b, c;
path auxp;
numeric i, auxn, yfact, sinfact, res;
pair cutp, vA, vB;
yfact = 30;
sinfact = 40;
res = 15;
a = pencircle scaled 3pt;
b = pencircle scaled 5pt;
c = pencircle scaled 1pt;
z1 = (1u,1u);
z2 = (4u,4u);
z3 = (4u,5u);
z4 = (3u,5u);
z5 = (3u,6u);
z6 = (4u,7u);
z7 = (6u,2.8u);
path cp;
cp = z1{up}..z2..z3..z4..z5..{up}z6;
draw z7 withpen b withcolor green;
draw cp withpen a;
auxp = hide( vA := unitvector(point 0 of cp - z7);
vB := unitvector(direction 0 of cp); )
( sinfact*((xpart vA)*(ypart vB) - (xpart vB)*(ypart vA)), 0 )
for i=1/res step 1/res until length cp:
hide( vA := unitvector(point i of cp - z7);
vB := unitvector(direction i of cp); )
...(sinfact*((xpart vA)*(ypart vB)-(xpart vB)*(ypart vA)),i*yfact)
endfor;
draw auxp withcolor blue+green;
draw origin--(sinfact,0) withcolor red;
draw origin--( 0, yfact*length cp ) withcolor red+green;
cutp = auxp intersectionpoint ( origin--( 0, yfact*length cp ) );
draw cutp withpen a;
auxn = ( ypart cutp )/yfact;
show auxn;
z8 = point auxn of cp;
draw z8 withpen b withcolor green;
draw z7--1.8[z7,z8] withpen c withcolor blue;
endfig;

end.
--------------------------------------------------------------------

It takes a search over the graph of the sine of the angle between the
line, connecting point z and a point on the path p, and the tangent of the
path p on that point. After the graph construction one finds the solution
point by intersecting the graph with a line going through the origin.
The sines are calculated with a crossproduct. Sines work better then
angles. Figure 0 was debugged using figure 1 where you can see the graph.

Improvements?

Lu\'{\i}s Nobre Gon\c{c}alves - http://matagalatlante.org

```