Tangent plane

The tangent plane to a smooth surface at a point $P$ is the plane that best approximates the surface near $P$. It’s the 2D analog of the tangent line to a 1D curve: the local linear approximation that touches the surface at $P$ and matches its first-order behaviour.

A smooth surface in 3D has a unique tangent plane at every point (and a unique unit normal direction, up to sign). Together they’re the surface’s local geometry, the flat ground a tiny insect would feel underfoot.

Two ways to specify a plane in 3D

A plane through point ${\mathbf{P}}_0=(x_0,y_0,z_0)$ is determined by either:

• A normal vector $\mathbf{n}$. Equation: $\mathbf{n}\cdot (\mathbf{r}-{\mathbf{P}}_0)=0$.
• Two non-parallel direction vectors $\mathbf{u},\mathbf{v}$. Parameterisation: $\mathbf{r}(s,t)={\mathbf{P}}_0+s\mathbf{u}+t\mathbf{v}$.

Both forms come up below. Sometimes the surface description gives you the normal directly, sometimes the tangent vectors.

From a parameterised surface

For a parameterised surface $\mathbf{r}(u,v)=⟨x(u,v),y(u,v),z(u,v)⟩$ at parameter point $(u_0,v_0)$:

The two partial derivatives

${\mathbf{r}}_u(u_0,v_0),{\mathbf{r}}_v(u_0,v_0)$

are tangent vectors to the $u$- and $v$-coordinate curves through $P$. They span the tangent plane.

The Cross product

$\mathbf{n}={\mathbf{r}}_u\times {\mathbf{r}}_v$

is normal to the tangent plane. The plane equation:

$\mathbf{n}\cdot (\mathbf{r}-{\mathbf{P}}_0)=0$

with ${\mathbf{P}}_0=\mathbf{r}(u_0,v_0)$.

The smoothness condition ${\mathbf{r}}_u\times {\mathbf{r}}_v\ne 0$ is exactly what guarantees the tangent plane is well-defined: nonzero cross product means ${\mathbf{r}}_u$ and ${\mathbf{r}}_v$ aren’t parallel, so they span a plane (not a line).

From a graph $z=f(x,y)$

For a surface given as the graph of a function, parameterise as $\mathbf{r}(x,y)=⟨x,y,f(x,y)⟩$. Then ${\mathbf{r}}_x=⟨1,0,f_x⟩$, ${\mathbf{r}}_y=⟨0,1,f_y⟩$, and

$\mathbf{n}={\mathbf{r}}_x\times {\mathbf{r}}_y=⟨-f_x,-f_y,1⟩.$

Tangent plane at $(x_0,y_0,f(x_0,y_0))$:

$z=f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0).$

This is the first-order Taylor expansion of $f$ at $(x_0,y_0)$. The tangent plane is the linear approximation $L(x,y)\approx f(x,y)$ for $(x,y)$ near $(x_0,y_0)$.

Worked example. $z=x^2+y^2$ at $(1,1,2)$. $f_x=2x=2$, $f_y=2y=2$. Tangent plane:

$z=2+2(x-1)+2(y-1)=2x+2y-2.$

Check: at $(1,1)$, the plane gives $z=2+2-2=2$. Match.

From a level surface $F(x,y,z)=c$

For a surface defined implicitly by $F(x,y,z)=c$, the gradient $\nabla F$ is normal to the surface (because the gradient is perpendicular to level sets). So:

$\mathbf{n}=\nabla F(x_0,y_0,z_0).$

Tangent plane:

$\nabla F({\mathbf{P}}_0)\cdot (\mathbf{r}-{\mathbf{P}}_0)=0.$

This is often the cleanest approach when the surface is given implicitly — no need to convert to a parameterisation or solve for $z$.

Worked example. Sphere $x^2+y^2+z^2=14$ at $(1,2,3)$. $\nabla F=⟨2x,2y,2z⟩=⟨2,4,6⟩$. Tangent plane:

$2(x-1)+4(y-2)+6(z-3)=0⟺x+2y+3z=14.$

$⟨1,2,3⟩$ is itself the (unnormalised) outward normal to the sphere at $(1,2,3)$, since the sphere is centered at the origin, so the gradient points the same direction.

Where it’s useful

Being the local linear approximation, it shows up wherever first-order behaviour matters:

• Linear approximation of multivariable functions. $f(x,y)\approx L(x,y)$ where $L$ is the tangent plane equation. Used for error estimation and derivative-based optimization in 2D.
• Setting up surface flux integrals. The vector area element $d\mathbf{S}=\mathbf{n}dS$ uses the tangent-plane normal.
• Newton’s method in 2D. Iterate using the tangent plane as a local linearisation of the residual surface.
• Computer graphics shading. Surface normals (taken from tangent planes) determine how light reflects off polygonal models.
• Differential geometry foundations. The tangent plane at $P$ is the tangent space $T_{P}S$; the family of tangent spaces over all $P$ is the tangent bundle.
• Optimization on surfaces. Constrained optimization (Lagrange multipliers) characterises critical points by gradients lying in the surface’s normal direction, i.e. orthogonal to the tangent plane.

Versus tangent line

Side by side with the 1D case:

Concept	1D (tangent line)	2D (tangent plane)
Underlying object	Curve $\mathbf{r}(t)$	Surface $\mathbf{r}(u,v)$
Tangent direction(s)	One: ${\mathbf{r}}^{\prime }(t_0)$	Two: ${\mathbf{r}}_u,{\mathbf{r}}_v$
Normal direction(s)	Two (or any in the perpendicular plane)	One (up to sign)
Equation form	$\mathbf{r}(s)={\mathbf{P}}_0+s{\mathbf{r}}^{\prime }(t_0)$	$\mathbf{n}\cdot (\mathbf{r}-{\mathbf{P}}_0)=0$
First-order Taylor of	$f(t)$	$f(x,y)$

Same idea either way: one piece of linear geometry holding all the local first-order information.

In context

The three constructions above feed forward into Scalar surface integral, Flux integral, and the right-hand side of Stokes’ theorem / Divergence theorem.