Tangent Plane Calculator

Find the equation of the tangent plane to a surface at a given point, calculate normal vectors, and visualize the geometry.

Surface and Point Configuration

Explicit Surface z = f(x,y)

Use * for multiplication, ^ for exponents

Current Surface:

Enter surface equation to see it displayed here

3D Visualization

About Tangent Planes

What is a Tangent Plane?

A tangent plane to a surface at a given point is the plane that best approximates the surface near that point. It contains all tangent lines to curves on the surface passing through the point. The tangent plane is fundamental in multivariable calculus, differential geometry, and optimization.

Surface Types

Explicit Surface

z = f(x,y)

Surface defined as a function of two variables. Examples: z = x² + y², z = sin(x)cos(y)

Implicit Surface

F(x,y,z) = 0

Surface defined by a level set. Examples: x² + y² + z² = 4, x² + y² - z² = 1

Parametric Surface

r(u,v) = (x(u,v), y(u,v), z(u,v))

Surface defined parametrically. Examples: spheres, tori, complex surfaces

How to Use This Calculator

  1. Choose the surface type (explicit, implicit, or parametric)
  2. Enter the surface equation or parametric functions
  3. Specify the point where you want the tangent plane
  4. Click "Visualize Surface" to see the 3D representation
  5. Click "Calculate Tangent Plane" to get the complete analysis
  6. View the tangent plane equation, normal vector, and geometric properties

Mathematical Formulas

Explicit Surface (z = f(x,y)):
Implicit Surface (F(x,y,z) = 0):
Parametric Surface:

Input Examples

Explicit Examples

  • x^2 + y^2 (paraboloid)
  • sin(x)*cos(y) (wave surface)
  • sqrt(x^2 + y^2) (cone)
  • exp(-(x^2 + y^2)) (Gaussian)

Implicit Examples

  • x^2 + y^2 + z^2 - 4 (sphere)
  • x^2 + y^2 - z^2 (hyperboloid)
  • x^2/4 + y^2/9 + z^2/16 - 1 (ellipsoid)
  • x^2 + y^2 - z (paraboloid)

Applications

  • Optimization: Finding critical points and analyzing local behavior
  • Physics: Analyzing force fields and potential surfaces
  • Engineering: Surface analysis in CAD and manufacturing
  • Computer Graphics: Surface rendering and lighting calculations
  • Differential Geometry: Studying surface curvature and properties
  • Machine Learning: Understanding loss function landscapes