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

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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 

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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

elliptical paraboloid

ellipsoid

hyperboloid of one-sheet

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/