Hallo David,
> In the second image in this file, I am disappointed to find that the
> vector doesn't look tangent to the path at the tip of the position
> vector.
[...]
OK, there are several issues here, that might be confusing.
> %initialize scale
> numeric u; 6u=3in;
>
> %draw coordinate axes
[...]
> %calculate the velocity vector
> numeric t;
> t:=directiontime (1u,2u) of p;
wrong: directiontime looks at the path p and tries to find a slope of
(1u,2u). Since it cannot find one, t gets -1 (check with show t).
> pair d;
> d:=direction t of p;
since t = -1 it looks at the first point and finds the direction there.
> drawarrow ((0,0)--d) scaled u shifted (1u,2u);
This is messy, since d has the correct correct direction but a "somewhat
arbitrary" magnitude. See the Metapost manual p. 28.
> p:=p scaled u;
If you do some calculations with path p, you have to scale it before doing
these calculations!
So here is my solution (tested with context/metafun)
----------------------------------
%initialize scale
numeric u; 6u=3in;
%draw coordinate axes
drawdblarrow (-1u,0)--(5u,0);
label.rt(btex $x$ etex, (5u,0));
drawdblarrow (0,-1u)--(0,3u);
label.top(btex $y$ etex, (0,3u));
%path
path p;
p:=(-1,1)...(1,2)...(5,2.5);
p:=p scaled u;
%position vector
drawarrow (0,0)--(1u,2u) withcolor red;
label.lrt(btex ${\bf r}(t)$ etex, 0.5[(0,0),(1u,2u)]);
dotlabel.ulft(btex $(x(t),y(t))$ etex, (1u,2u));
%calculate the velocity vector
numeric t;
% at what time does my point intersect w/ the path?
(t,whatever) = p intersectiontimes ((1u,-infinity)--(1u,infinity));
pair d;
d:=direction t of p;
%warning! d now has the correct direction but not!! magnitude
drawarrow (origin -- ((2u,0) rotated angle d))
shifted (1u,2u)
withcolor green;
draw p withcolor blue;
---------------------------------------
--
Viele Gr��e,
Patrick Gundlach
