Conics sections in 3D world

Conic sections in real-life

How they appear in 3 dimensional space

A conic section is a curve obtained from a cone's surface intersecting a plane.

​

Conic section in 2D space

Demonstration in Sketchup

Home

Open in new tab

Copy link

Reflection property of parabola

A short experiment

​Reflection property of parabola

Reminder:
any point on the parabola is equidistant from the directrix and the focus.

We are living in a
3 dimensional world

We are going to discuss the 3D counterparts to our 2D conic sections.

Quadric surfaces & Cylinder

Cylinder

Parabolic/Hyperbolic Cylinder

Demonstration in Sketchup
- Cylinder

Sphere

Ellipsoid

Demonstration in Sketchup
- Sphere

Demonstration in Sketchup
- Elliptical Paraboloid

Elliptical paraboloid
in real-life

• Please search the internet for anything or any structure that is in an elliptical paraboloid shape.

• Send me as quick as possible by AirDrop.

Radio telescope

Reflector installed in the lighthouse

is traditionally lit at Olympia, Greece, using a parabolic reflector concentrating sunlight

​

The Olympic Flame

Demonstration by Play-Doh
- Hyperbolic Paraboloid

Hyperbolic paraboloid
in real-life

• Raise your hands and answer directly

Lee Valley VeloPark, Stratford, London

​​The James Evans house of 1960

Demonstration in Sketchup
- Hyperboloid

Hyperboloid in real-life

• Please search the internet for anything or any structure that is in a hyperboloid shape.

• Either one-sheet or double-sheet

• Send me as quick as possible by AirDrop.

The Corporation Street Bridge in Manchester, England

Cooling tower of a nuclear power plant

Canton Tower, Guangzhou, China

National Stadium Singapore

​​Sonic boom

shock waves created when an object travels through the air faster than the speed of sound

Further Discussion
- equation of
elliptical paraboloid 

Home

Open in new tab

Copy link

Concept Check
 

Labelling

Match the names with the corresponding surfaces shown in the figure.

Drag labels to their correct position on the image

hyperboloid of two-sheet

hyperbolic paraboloid

hyperboloid of one-sheet

elliptical paraboloid

ellipsoid

Multiple Choice

Every plane section of a paraboloid by a plane parallel to the axis of symmetry is a...

• circle

• ellipse

• parabola

• hyperbola

Multiple Choice

Every plane section of a hyperboloid by a plane parallel to the axis of symmetry is a...

• circle

• ellipse

• parabola

• hyperbola

Multiple Choice

Any other plane section of a elliptical paraboloid is a...

• circle

• ellipse

• parabola

• hyperbola

Multiple Choice

Any other plane section of a hyperbolic paraboloid is a...

• circle

• ellipse

• parabola

• hyperbola

Summary
 

• ​Conic sections are the plane section of the quadratic surfaces (e.g. cones) in 3 dimensional space

• The 3D surfaces can be obtained by rotating or translating the 2D conic sections along the axis of symmetry.

​More reference

https://math.libretexts.org/Bookshelves/Calculus/CLP-4_Vector_Calculus_(Feldman_Rechnitzer_and_Yeager)/06%3A_Appendices/6.01%3A_A_Appendices/6.1.08%3A_A.8_Conic_Sections_and_Quadric_Surfaces

​More reference

https://calcworkshop.com/vectors-and-the-geometry-of-space/quadric-surfaces/