Calculate Tangent Plane to Surface

This example shows how to approximate gradients of a function by finite differences. It then shows how to plot a tangent plane to a point on the surface by using these approximated gradients.

Create the function $$f(x,y)=x^2+y^2$$ using a function handle.

f = @(x,y) x.^2 + y.^2;

Approximate the partial derivatives of $$f(x,y)$$ with respect to $$x$$ and $$y$$ by using the gradient function. Choose a finite difference length that is the same as the mesh size.

[xx,yy] = meshgrid(-5:0.25:5);
[fx,fy] = gradient(f(xx,yy),0.25);

The tangent plane to a point on the surface, $$P=(x_0,y_0,f(x_0,y_0))$$, is given by

$$z=f(x_0,y_0)+\frac{\partial f(x_0,y_0)}{\partial x}(x-x_0)+\frac{\partial f(x_0,y_0)}{\partial y}(y-y_0).$$

The fx and fy matrices are approximations to the partial derivatives $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\partial y}$$. The point of interest in this example, where the tangent plane meets the functional surface, is (x0,y0) = (1,2). The function value at this point of interest is f(1,2) = 5.

To approximate the tangent plane z you need to find the value of the derivatives at the point of interest. Obtain the index of that point, and find the approximate derivatives there.

x0 = 1;
y0 = 2;
t = (xx == x0) & (yy == y0);
indt = find(t);
fx0 = fx(indt);
fy0 = fy(indt);

Create a function handle with the equation of the tangent plane z.

z = @(x,y) f(x0,y0) + fx0*(x-x0) + fy0*(y-y0);

Plot the original function $$f(x,y)$$, the point P, and a piece of plane z that is tangent to the function at P.

surf(xx,yy,f(xx,yy),'EdgeAlpha',0.7,'FaceAlpha',0.9)
hold on
surf(xx,yy,z(xx,yy))
plot3(1,2,f(1,2),'r*')

View a side profile.

view(-135,9)