Fit a curve in MATLAB where points have specified normals - matlab

Question

I have two 2D points p1, p2 in MATLAB, and each point has a normal n1, n2. I wish to find the (cubic) polynomial which joins the two points and agrees with the specified normals at each end. Is there something built-in to MATLAB to do this?
Of course, I could derive the equations for the polynomial manually, but MATLAB's curve fitting toolbox has so much built-in that I assumed it would be possible. I haven't been able to find any examples of curve, spline or polynomial fitting where the normals are specified.
As an extrapolation of this, I would like to fit splines where each data point has a normal specified.

Answer 1

1. If your points are points of a function, then you need cubic Hermite spline interpolation:
In numerical analysis, a cubic Hermite spline or cubic Hermite
interpolator is a spline where each piece is a third-degree polynomial
specified in Hermite form: that is, by its values and first derivatives at the
end points of the corresponding domain interval.
Cubic Hermite splines
are typically used for interpolation of numeric data specified at
given argument values x(1), x(2), ..., x(n), to obtain a smooth
continuous function. The data should consist of the desired function
value and derivative at each x(k). (If only the values are provided,
the derivatives must be estimated from them.) The Hermite formula is
applied to each interval (x(k), x(k+1)) separately. The resulting
spline will be continuous and will have continuous first derivative.
Cubic polynomial splines can be specified in other ways, the Bézier
form being the most common. However, these two methods provide the
same set of splines, and data can be easily converted between the
Bézier and Hermite forms; so the names are often used as if they were
synonymous.
Specifying the normals at each point is the same as specifying the tangents (slopes, 1st derivatives), because the latter are perpendicular to the former.
In Matlab, the function for calculating the Piecewise Cubic Hermite Interpolating Polynomial is pchip. The only problem is that pchip is a bit too clever:
The careful reader will notice that pchip takes function values as
input, but no derivative values. This is because pchip uses the
function values f(x) to estimate the derivative values. [...] To do a
good derivative approximation, the function has to use an
approximation using 4 or more points [...] Luckily, using Matlab we
can write our own functions to do interpolation using real cubic
Hermite splines.
...the author shows how to do this, using the function mkpp.
2. If your points are not necessarily points of a function, then each interval should be interpolated by a quadratic Bezier curve:
In this example, 3 points are given: the endpoints P(0) and P(2), and P(1), which is the intersection of the tangents at the endpoints. The position of P(1) can be easily calculated from the coordinates of P(0) and P(2), and the normals at these points.
In Matlab, you can use spmak, see the examples here and here.

Answer 2

You could do something like this:
function neumann_spline(p, m, q, n)
% example data
p = [0; 1];
q = [2; 5];
m = [0; 1];
n = [1; 1];
if (m(2) ~= 0)
s1 = atan(-m(1)/m(2));
else
s1 = pi/2;
end
if (n(2) ~= 0)
s2 = atan(-n(1)/n(2));
else
s2 = pi/2;
end
hold on
grid on
axis equal
plot([p(1) p(1)+0.5*m(1)], [p(2) p(2)+0.5*m(2)], 'r', 'Linewidth', 1)
plot([q(1) q(1)+0.5*n(1)], [q(2) q(2)+0.5*n(2)], 'r', 'Linewidth', 1)
sp = csape([p(1) q(1)], [s1 p(2) q(2) s2], [1 1]);
fnplt(sp)
plot(p(1), p(2), 'k.', 'MarkerSize', 16)
plot(q(1), q(2), 'k.', 'MarkerSize', 16)
title('Cubic spline with prescribed normals at the endpoints')
end
The result is