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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)=x2+y2 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=(x0,y0,f(x0,y0)), is given by

z=f(x0,y0)+f(x0,y0)x(x-x0)+f(x0,y0)y(y-y0).

The fx and fy matrices are approximations to the partial derivatives fx and fy. 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*')

Figure contains an axes. The axes contains 3 objects of type surface, line.

View a side profile.

view(-135,9)

Figure contains an axes. The axes contains 3 objects of type surface, line.

See Also

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