# Calculate perpendicular line to an 3rd order polynomial

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Edited: darova on 25 May 2021
Hi all,
I have been trying to calculate the perpendicular line to an 3rd order (or higher) polynomial line.
I found a way to do it with an 2nd order polynomial, however, can't work out how to do it for any higher.
This calcualtion would make my data analysis a lot quicker, but is not essential. I've tried a bunch of stuff and I think this is my last hope.
Thanks in advance for any help.

Torsten on 24 May 2021
For a general function f, the formula for the perpendicular line to the function through (x0,f(x0)) is
y = f(x0) + (x-x0)* ( -1/f'(x0) )
Thanks mate.
Helped me figure it out.
For people looking for something similar I have written this basic script that should help.
It could probably be done better and more efficient, but I am a bit of a Matlab rookie.
Hopefully this is a good start for anyone looking. See below.
pow = 4; % power of the poly can be between 3 and 5.
% It could be more, but it is constrained by the if function in the script.
x = linspace(0,1,5);
y = 1./(1+x);
scatter(x,y);
p = polyfit(x,y,pow);
hold on
yy = polyval(p,x);
plot(x,yy)
k2 = polyder(p);
if pow==3
y2 = k2(1)*x.^2 + k2(2)*x + k2(3);
elseif pow==4
y2 = k2(1)*x.^3 + k2(2)*x.^2 + k2(3)*x + k2(4);
elseif pow==5
y2 = k2(1)*x.^4 + k2(2)*x.^3 + k2(3)*x.^2 + k2(4)*x + k2(5);
end
for i = 2:4
y3 = y2(i);
pe = -1/y3;
X1 = x(i);
Y1 = y(i);
X2 = X1+.2;
Y2 = pe*(X2 - X1)+Y1;
plot([X2, X1], [Y2, Y1],':','LineWidth', 2);
X3 = X1-.2;
Y3 = pe*(X3 - X1)+Y1;
plot([X3, X1], [Y3, Y1],'--', 'LineWidth', 2);
end
axis equal

R2021a

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