Tangent Plane: Interactive 3D Visualization

01

Tangent plane in this visualization

If $f(x, y)$ is differentiable at $(a, b)$ , the surface $z = f(x, y)$ has a tangent plane there: the graph of the linear approximation, or linearization, $L$ . It is built from the two partial derivatives, and it is the one plane that the surface hugs ever more closely as you approach the point. The point $(a, b)$ is also written $(x_0, y_0)$ .

Tangent plane

z = L(x, y) = f(a, b) + f_x(a, b)\,(x - a) + f_y(a, b)\,(y - b)

The glass surface is the graph of f, the blue sheet is the plane at P, and the blue lines in it are the tangents to the two slices of the surface through P (along x and along y): they span the plane. The red square on the map is $|x - a| \le w,\ |y - b| \le w$ . Shrinking w magnifies the view around P, by the same factor in x, y and z. The yellow segment marks the largest vertical gap between surface and plane over the square (the part inside the domain, estimated on a fine grid); the readout divides it by w. Being differentiable means exactly that this ratio tends to 0, that is, the largest gap is o(w):

Differentiable at (a, b)

\max_{|x-a|,\,|y-b| \le w} \frac{|f(x, y) - L(x, y)|}{w} \to 0 \quad (w \to 0)

02

Things to notice

Where the second-order part of f at P does not vanish (the hill, the waves at most points), the gap shrinks like $w^2$ once w is small, so each tenfold zoom makes the ratio about ten times smaller; on the bowl and the saddle it is exactly so at every point. Where the second-order part vanishes it shrinks faster: at the origin of the waves, a hundredfold per tenfold zoom.
Partial derivatives are not enough. On the fold (this page’s name for a classic counterexample $f = 0.9\,xy/\sqrt{x^2 + y^2}$ , $f(0, 0) = 0$ ) both are 0 at the origin and give the plane $z = 0$ , but the largest gap stays $(0.9/\sqrt2)\,w \approx 0.636\,w$ , at the square’s four corners, at every zoom: no tangent plane there. At the tip of the cone the partial derivatives do not even exist. Everywhere else both surfaces are differentiable.
A tangent plane may cross the surface: at a saddle point it is horizontal, with the surface above it in one direction and below in another.
The 3D view is drawn in perspective, so tilts on screen depend on the viewpoint. The readout shows the plane’s equation with its numbers rounded (≈), and the largest gap is a sampled estimate, not a bound.

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Where it is used

Linear approximation: near $(a, b)$ , $f(x, y) \approx L(x, y)$ ; the change $dz = f_x\,dx + f_y\,dy$ is the total differential. The vector $(f_x, f_y, -1)$ is normal to the plane.
In numerical methods, local derivative information builds each step of an iteration: Newton’s method for a system of equations solves the linearized equations, and gradient descent follows the gradient, which is the tilt of the tangent plane.

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Controls

Slide the fader toward 0.001 to zoom in; w → 0 zooms in a decade at a time. On the whole-surface view (w = 1) drag P on the surface or its foot on the map; the wheels and key points move it at any zoom. Slices shows the two slices and their tangents; Map shows where you are.
Presentation mode opens a full-screen view for projectors (Space zooms in, ← → change w tenfold, H hides the answers).

05

Tangent plane in this visualization

Things to notice

Where it is used

Controls

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