Multivariable Calculus: 3D Quadric Surfaces, Implicit Equations & Interactive WebGL Geometry

In multivariable calculus and computer graphics, 3D quadric surfaces are continuous 2D manifolds embedded within 3D Euclidean space $\mathbb{R}^3$ . They are defined implicitly by the general second-degree equation in three variables $(x, y, z)$ :

A x^2 + B y^2 + C z^2 + D xy + E yz + F xz + G x + H y + I z + J = 0

Depending on the signs and values of coefficients $A, B, C$ , quadric surfaces take the form of spheres, ellipsoids, paraboloids, hyperboloids, or tori rings.

Series·Interactive Multivariable Calculus & Geometry

Chapter 3 of 5

1Geometric Mastery: Circle Equations, Parametric Geometry & Interactive Proofs

Read →

2Calculus & Geometry: 2D Function Analysis, Tangent Slopes & Interactive Curve Plotting

Read →

3Multivariable Calculus: 3D Quadric Surfaces, Implicit Equations & Interactive WebGL Geometry

Current

4Advanced Calculus: Rigorous Integration Theory, Special Forms & Numerical Algorithms

Read →

5Step-by-Step Calculus Solutions: Three Classic Definite & Indefinite Integration Problems

Read →

Calculus & Geometry: 2D Function Analysis, Tangent Slopes & Interactive Curve Plotting

Advanced Calculus: Rigorous Integration Theory, Special Forms & Numerical Algorithms

1. Interactive 3D Surface Visualizer

Below is an interactive Three.js WebGL 3D Graph Engine. Use the dropdown preset selector to switch between quadric equations ( $x^2+y^2+z^2=R^2$ , Paraboloid, Hyperboloid, Torus), toggle wireframe mesh mode, or drag with your mouse/touch to orbit in 3D space:

Interactive 3D Quadric Surface & Implicit Graph Engine

z = implicit surface: f(xz = yz = z) = 0

3D Surfaces:

Drag to rotate • Scroll to zoom

Hover surface to inspect (x, y, z)

2. Classification of Quadric Surfaces in $\mathbb{R}^3$

The 3D Sphere ( $x^2 + y^2 + z^2 = R^2$ )

The most symmetric 3D quadric surface. Every point on the boundary maintains distance $R$ from center $(0, 0, 0)$ :

x^2 + y^2 + z^2 = R^2

Parametric surface equations in spherical coordinates $(\theta, \phi)$ :

\begin{cases} x(\theta, \phi) = R \sin\phi \cos\theta \\ y(\theta, \phi) = R \sin\phi \sin\theta \\ z(\theta, \phi) = R \cos\phi \end{cases} \quad \text{where } \theta \in [0, 2\pi), \; \phi \in [0, \pi]

Elliptic Paraboloid ( $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ )

A bowl-shaped surface where cross-sections parallel to the $XY$ -plane are ellipses, and cross-sections parallel to the $XZ$ - and $YZ$ -planes are parabolas:

z = x^2 + y^2

Hyperbolic Paraboloid / Saddle Surface ( $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ )

A doubly ruled surface resembling a horse saddle. It exhibits a local minimax point (saddle point) at the origin $(0, 0, 0)$ :

z = x^2 - y^2

Hyperboloid of One Sheet ( $x^2 + y^2 - z^2 = 1$ )

A continuous hourglass shape used in structural cooling towers and hyperboloid architecture:

x^2 + y^2 - z^2 = 1

3. Surface Area Integrals & Differential Elements

The surface area $A$ of an explicit function $z = f(x, y)$ over domain $D \subset \mathbb{R}^2$ is computed via double integration:

A = \iint_{D} \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dx \, dy

For the sphere $z = \sqrt{R^2 - x^2 - y^2}$ , transforming to polar coordinates verifies Archimedes' famous surface area formula:

A = 4\pi R^2

4. Quadric Surface Comparison Reference

Surface Name	Implicit Equation	Cross-Section Traces
Sphere	$x^2 + y^2 + z^2 = R^2$	Circles in all principal planes
Elliptic Paraboloid	$z = x^2 + y^2$	Ellipses for $z > 0$ , Parabolas in $XZ/YZ$
Hyperbolic Paraboloid	$z = x^2 - y^2$	Hyperbolas for $z \neq 0$ , Parabolas in $XZ/YZ$
Hyperboloid (1 Sheet)	$x^2 + y^2 - z^2 = 1$	Circles parallel to $XY$ , Hyperbolas in $XZ/YZ$
Torus Ring	$(R - \sqrt{x^2+y^2})^2 + z^2 = r^2$	Concentric circles & Cassini ovals

A x^2 + B y^2 + C z^2 + D xy + E yz + F xz + G x + H y + I z + J = 0

Depending on the signs and values of coefficients $A, B, C$ , quadric surfaces take the form of spheres, ellipsoids, paraboloids, hyperboloids, or tori rings.

Series·Interactive Multivariable Calculus & Geometry

Chapter 3 of 5

1Geometric Mastery: Circle Equations, Parametric Geometry & Interactive Proofs

Read →

2Calculus & Geometry: 2D Function Analysis, Tangent Slopes & Interactive Curve Plotting

Read →

3Multivariable Calculus: 3D Quadric Surfaces, Implicit Equations & Interactive WebGL Geometry

Current

4Advanced Calculus: Rigorous Integration Theory, Special Forms & Numerical Algorithms

Read →

5Step-by-Step Calculus Solutions: Three Classic Definite & Indefinite Integration Problems

Read →

Calculus & Geometry: 2D Function Analysis, Tangent Slopes & Interactive Curve Plotting

Advanced Calculus: Rigorous Integration Theory, Special Forms & Numerical Algorithms

1. Interactive 3D Surface Visualizer

Interactive 3D Quadric Surface & Implicit Graph Engine

z = implicit surface: f(xz = yz = z) = 0

3D Surfaces:

Drag to rotate • Scroll to zoom

Hover surface to inspect (x, y, z)

2. Classification of Quadric Surfaces in $\mathbb{R}^3$

The 3D Sphere ( $x^2 + y^2 + z^2 = R^2$ )

The most symmetric 3D quadric surface. Every point on the boundary maintains distance $R$ from center $(0, 0, 0)$ :

x^2 + y^2 + z^2 = R^2

Parametric surface equations in spherical coordinates $(\theta, \phi)$ :

\begin{cases} x(\theta, \phi) = R \sin\phi \cos\theta \\ y(\theta, \phi) = R \sin\phi \sin\theta \\ z(\theta, \phi) = R \cos\phi \end{cases} \quad \text{where } \theta \in [0, 2\pi), \; \phi \in [0, \pi]

Elliptic Paraboloid ( $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ )

A bowl-shaped surface where cross-sections parallel to the $XY$ -plane are ellipses, and cross-sections parallel to the $XZ$ - and $YZ$ -planes are parabolas:

z = x^2 + y^2

Hyperbolic Paraboloid / Saddle Surface ( $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ )

A doubly ruled surface resembling a horse saddle. It exhibits a local minimax point (saddle point) at the origin $(0, 0, 0)$ :

z = x^2 - y^2

Hyperboloid of One Sheet ( $x^2 + y^2 - z^2 = 1$ )

A continuous hourglass shape used in structural cooling towers and hyperboloid architecture:

x^2 + y^2 - z^2 = 1

3. Surface Area Integrals & Differential Elements

The surface area $A$ of an explicit function $z = f(x, y)$ over domain $D \subset \mathbb{R}^2$ is computed via double integration:

A = \iint_{D} \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} \, dx \, dy

For the sphere $z = \sqrt{R^2 - x^2 - y^2}$ , transforming to polar coordinates verifies Archimedes' famous surface area formula:

A = 4\pi R^2

4. Quadric Surface Comparison Reference

Surface Name	Implicit Equation	Cross-Section Traces
Sphere	$x^2 + y^2 + z^2 = R^2$	Circles in all principal planes
Elliptic Paraboloid	$z = x^2 + y^2$	Ellipses for $z > 0$ , Parabolas in $XZ/YZ$
Hyperbolic Paraboloid	$z = x^2 - y^2$	Hyperbolas for $z \neq 0$ , Parabolas in $XZ/YZ$
Hyperboloid (1 Sheet)	$x^2 + y^2 - z^2 = 1$	Circles parallel to $XY$ , Hyperbolas in $XZ/YZ$
Torus Ring	$(R - \sqrt{x^2+y^2})^2 + z^2 = r^2$	Concentric circles & Cassini ovals

1. Interactive 3D Surface Visualizer

2. Classification of Quadric Surfaces in R3\mathbb{R}^3R3

The 3D Sphere (x2+y2+z2=R2x^2 + y^2 + z^2 = R^2x2+y2+z2=R2)

Elliptic Paraboloid (z=x2a2+y2b2z = \frac{x^2}{a^2} + \frac{y^2}{b^2}z=a2x2​+b2y2​)

Hyperbolic Paraboloid / Saddle Surface (z=x2a2−y2b2z = \frac{x^2}{a^2} - \frac{y^2}{b^2}z=a2x2​−b2y2​)

Hyperboloid of One Sheet (x2+y2−z2=1x^2 + y^2 - z^2 = 1x2+y2−z2=1)

3. Surface Area Integrals & Differential Elements

4. Quadric Surface Comparison Reference

1. Interactive 3D Surface Visualizer

2. Classification of Quadric Surfaces in R3\mathbb{R}^3R3

The 3D Sphere (x2+y2+z2=R2x^2 + y^2 + z^2 = R^2x2+y2+z2=R2)

Elliptic Paraboloid (z=x2a2+y2b2z = \frac{x^2}{a^2} + \frac{y^2}{b^2}z=a2x2​+b2y2​)

Hyperbolic Paraboloid / Saddle Surface (z=x2a2−y2b2z = \frac{x^2}{a^2} - \frac{y^2}{b^2}z=a2x2​−b2y2​)

Hyperboloid of One Sheet (x2+y2−z2=1x^2 + y^2 - z^2 = 1x2+y2−z2=1)

3. Surface Area Integrals & Differential Elements

4. Quadric Surface Comparison Reference

2. Classification of Quadric Surfaces in $\mathbb{R}^3$

The 3D Sphere ( $x^2 + y^2 + z^2 = R^2$ )

Elliptic Paraboloid ( $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ )

Hyperbolic Paraboloid / Saddle Surface ( $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ )

Hyperboloid of One Sheet ( $x^2 + y^2 - z^2 = 1$ )

2. Classification of Quadric Surfaces in $\mathbb{R}^3$

The 3D Sphere ( $x^2 + y^2 + z^2 = R^2$ )

Elliptic Paraboloid ( $z = \frac{x^2}{a^2} + \frac{y^2}{b^2}$ )

Hyperbolic Paraboloid / Saddle Surface ( $z = \frac{x^2}{a^2} - \frac{y^2}{b^2}$ )

Hyperboloid of One Sheet ( $x^2 + y^2 - z^2 = 1$ )