| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
I'm drawing a logarithmic spiral as defined in polar coordinates by the
equation:
r = a * exp(t)
Where a is a constant, and r is the radius, t the angle. This is a
convenient parametric form and I can readily compute (x,y) coordinates
via the standard substitutions (x,y) = (r cos(t),r sin(t)).
Trouble is I also want to occasionally draw a tangent to the spiral
for a few (random) values of t. But since the equation is in polar form
(and not really even a function!) I don't remember how to go about
calculating the derivative.
Anybody happen to remember enough calculus to help out here?
John
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 1910.1 | SWAG | AUSSIE::GARSON | achtentachtig kacheltjes | Sun Nov 20 1994 21:13 | 14 |
re .0
I think you can *informally* deal with this using the following.
dy dy dx
-- = -- / --
dx dt dt
Substituting everything in should eventually give you dy/dx as a
function of t.
For your particular function I get dy/dx = (sin t + cos t)/(cos t - sin t)
and I think that probably simplifies but I can't remember my trig formulae.
dy/dx = tan(t+pi/4) ?
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| 1910.2 | WRKSYS::ROTH | Geometry is the real life! | Mon Nov 21 1994 15:44 | 19 | |
You could just take the velocity vector and normalize it to a unit
vector to get the tangent
P(t) = (a exp(t) cos(t), a exp(t) sin(t))
V(t) = (a exp(t) (cos(t) - sin(t)), a exp(t) (sin(t) + cos(t)))
V(t) doesn't vanish away from the origin so the x and y components can
be normalized.
You could even do something sleazy like make a small change up and
down in t and use central differences to approximate the derivative
to second order accuracy if you don't want to figure it out :-)
This example is trivial, but I've often had complicated functions
and have compared the numerical estimate with the analytic derivative
to check on mistakes in the latter.
- Jim
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| 1910.3 | AUSSIE::GARSON | achtentachtig kacheltjes | Mon Nov 21 1994 16:21 | 5 | |
additional to .1
.2 reminds me to observe that the velocity vector makes a constant
angle with the position vector (in this case pi/4) which I believe is
characteristic of a logarithmic spiral.
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