Reply sent privately (in error) on 24 August
Mico Filós wrote:
Hi,
I am sorry if this is a trivial question but I cannot figure out how
to do this.
Imagine I have a nonlinear function f(x) and its linearization at some
point (x0,f(x0)),
given by y(x) = f'(x0) (x - x0) + y0, where y0 = f(x0). Imagine also that
I don't use the same scale for the x and y axes. How can I obtain a
straight line that
*looks* perpendicular to the tangent y(x)?
Since the x and y scales are not the same, a line with slope equal to
-1/f'(x0)
does not look perpendicular to y(x). I want to plot a straight line
that passes through
an arbitrary point of the tangent (not necessarily (x0,y0)) and that
looks perpendicular to
it. My idea is actually to plot the projection of an arbitrary point
of the nonlinear function,
(x1,f(x1)) to the subspace spanned by the tangent at x0. For me the
most straightforward thing to do would be:
* Compute the angle theta (on the actual canvas, i.e., as it looks
on the screen) of the path
defined by the tangent function y(x)
* Plot a line path L with an angle theta + pi/2 that passes through
some particular point
of the tangent Q=(x0', y0'), which I specify.
* Find the intersection R of the line L with the path defined by
the nonlinear function.
* Ideally, I would stroke only the portion of L that connects the
point
(x0',y0') with the intersection point R.
Describing this in words is painful. I hope you see what I mean.
Thanks a lot in for your patience.
Mico
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------------------------------------------------------------------------
from pyx import *
g = graph.graphxy(width=8, x=graph.axis.linear(min=-5, max=5))
g.plot(graph.data.function("y(x)=x*x"))
# Let us plot the tangent at (x0,y0) = (2,4) slope 4
g.plot(graph.data.function("y(x)=4*(x-1)"))
"""
Let us take two more points on the tangent
"""
slope=4
deltaX=2
x1=-3 # example, you can change this
x2 = x1 + deltaX
y1 = slope * (x1 -1)
y2 = slope * (x2 - 1)
"""
Now you need to get the two points (x1,y1), (x2,y2)
into canvas-coordinates rather than scaled coordinates.
This is the crucial stage that will provide for your peculiar
request that the line *looks* perpendicular to the tangent line.
I shall call these U-V coordinates, and their origin is bottom left of the
canvas (g), their units the usual user-units (e.g cm)
"""
u1,v1= g.pos(x1,y1)
u2,v2= g.pos(x2,y2)
deltaU=u2-u1
deltaV=v2-v1
# lastly a point (u3,v3) that can define your 'normal' from (u1,v1)
u3=u1-deltaV
v3=v1+deltaU
# as these are in canvas coordinates, not graph ones, we use
# stroke .. path .. etc to plot a segment from u1,v1 to u3,v3
g.stroke(path.path(path.moveto(u1,v1),path.lineto(u3,v3)))
g.writePDFfile("graph3")
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