Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!

Elementary Surfaces

Ellipsoid, sphere, hyperboloid, cone and more.

Sponsored Links

The simplest of surfaces are the unbound planes. The next complexity are the second degree or quadric surfaces.

Non-degenerate Quadrics


x2/a2 + y2/b2 + z2/c2 = 1                       (1)


a, b, c = intrinsic parameters


Sphere, a special ellipsoid, can be expressed as

x2 + y2 + z2 = a2                            (2)


a = radius

Hyperboloid of one sheet

x2/a2 + y2/b2 - z2/c2 = 1                            (3)


a, b, c = intrinsic parameters

Hyperboloid of two sheets

x2/a2 + y2/b2 - z2/c2 = -1                            (4)


a, b, c = intrinsic parameters

Quadric Cone

x2/a2 + y2/b2 - z2/c2 = 0                             (5)


a, b, c = intrinsic parameters

Elliptic Paraboloid

x2/a2 + y2/b2 - 2 z = 0                                (6)


a, b = intrinsic parameters

Elliptic Cylinder

x2/a2 + y2/b2 = 1                                (7)


a, b = intrinsic parameters


x2 + y2 = a2                                (8)


a, b = intrinsic parameters

Hyperbolic Cylinder

x2/a2 - y2/b2 = 1                                 (9)


a, b = intrinsic parameters

Parabolic Cylinder

y2 - 2 l x = 0                             (10)


l = intrinsic parameters

Degenerate Quadrics

Parallel Planes

x2 - a2 = 0                             (11)


a = half the distance between the parallel planes

Intersecting Planes

x2/a2 - y2/b2 = 0                          (12)


tan-1(b/a) = half the angle between the intersecting planes

Coincident Planes

x2 = 0                            (13)

Sponsored Links

Related Topics

Related Documents

Sponsored Links

Engineering ToolBox - SketchUp Extension - Online 3D modeling!

3D Engineering ToolBox Extension to SketchUp - add parametric components to your SketchUp model

Add standard and customized parametric components - like flange beams, lumbers, piping, stairs and more - to your Sketchup model with the Engineering ToolBox - SketchUp Extension - enabled for use with the amazing, fun and free SketchUp Make and SketchUp Pro .Add the Engineering ToolBox extension to your SketchUp from the SketchUp Pro Sketchup Extension Warehouse!

About the Engineering ToolBox!


We don't collect information from our users. Only emails and answers are saved in our archive. Cookies are only used in the browser to improve user experience.

Some of our calculators and applications let you save application data to your local computer. These applications will - due to browser restrictions - send data between your browser and our server. We don't save this data.

Google use cookies for serving our ads and handling visitor statistics. Please read Google Privacy & Terms for more information about how you can control adserving and the information collected.

AddThis use cookies for handling links to social media. Please read AddThis Privacy for more information.


This page can be cited as

  • Engineering ToolBox, (2009). Elementary Surfaces. [online] Available at: [Accessed Day Mo. Year].

Modify access date.

. .


3D Engineering ToolBox - draw and model technical applications! 2D Engineering ToolBox - create and share online diagram drawing templates! Engineering ToolBox Apps - mobile online and offline engineering applications!

Scientific Online Calculator

Scientific Calculator

3 10

Sponsored Links