Tangent vector

Tangent vector

[David Arnold]

All,

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.

input mp-tool;

beginfig(1);

%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;
 draw p withcolor blue;

%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));
endfig;

beginfig(2);

%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);

%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;
 t:=directiontime (1u,2u) of p;
 pair d;
 d:=direction t of p;
 drawarrow ((0,0)--d) scaled u shifted (1u,2u);

 p:=p scaled u;
 draw p withcolor blue;

endfig;

end.

Re: Tangent vector

[Hans Hagen]

At 02:47 PM 9/26/2001 -0700, David Arnold wrote:

>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.

now, are your disappointed in me, metapost, or yourself -)

how about:  d:=direction 1 of p; since you *know* that 1,2 (not 1u,2u !) is 
point 1 there is no need for time calculations

also, keep in mind that this times/point stuff is sensitive to rounding 
errors (in that respect you can indeed be disappointed in mp)

now, your error result from calculating a time for a scaled point (watch 
the 1u,2u) while you have no scaled path but scale it afterwards.

i think that your code snippet will become more clean if you say  p := p 
scaled u after setting p and then work on the scaled path, or alternatively 
use no u's and only scale the whole to u, but this is a matter of taste.

i've fallen in this trap some times too [at first i thought that your 
sample was a rounding error; suggestion: in such case draw d withpen 
pencircle scaled 5 helps; add this to your code and you will see that you 
got the point + 1,2 near point zero which indeed has the tangent you got.

Hans
-------------------------------------------------------------------------
                                   Hans Hagen | PRAGMA ADE | pragma@wxs.nl
                       Ridderstraat 27 | 8061 GH Hasselt | The Netherlands
  tel: +31 (0)38 477 53 69 | fax: +31 (0)38 477 53 74 | www.pragma-ade.com
-------------------------------------------------------------------------

Tangent vector

[Patrick Gundlach]

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[-- Attachment #1: Type: text/plain; charset="us-ascii", Size: 1907 bytes --]

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 Grüße, 

    Patrick Gundlach

Re: Tangent vector

[Hans Hagen]

At 11:24 PM 9/29/2001 +0200, Patrick Gundlach wrote:

>  % at what time does my point intersect w/ the path?
>  (t,whatever) = p intersectiontimes ((1u,-infinity)--(1u,infinity));

ah, that's real metapost. i still have to learn to think in whatevers. 
maybe this also is less sensitive to rounding problems

Hans
-------------------------------------------------------------------------
                                   Hans Hagen | PRAGMA ADE | pragma@wxs.nl
                       Ridderstraat 27 | 8061 GH Hasselt | The Netherlands
  tel: +31 (0)38 477 53 69 | fax: +31 (0)38 477 53 74 | www.pragma-ade.com
-------------------------------------------------------------------------