It is study notes for conicoids. It is fundamental part of geometry., Study notes of Geometry

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Conicoids

Some definitions

• Let S denote the surface under consideration. The intersection of the

surface with a plane P is called a section of the surface and is obtained

by considering two equations simultaneously.

• The intersection of the surface with the 𝑥𝑦- plane is its 𝑥𝑦- trace and it

is obtained by equating 𝑧 = 0 in the equation for S.

• The nonzero point of intersection of the surface and a coordinate axis

is an intercept.

• Two points are said to be symmetric with respect to a plane if the line

segment joining them is bisected by the plane and is perpendicular to

the plane.

• A surface S is symmetric with respect to a plane P if, whenever a point

A lies on the surface S, the point B which is symmetric to A with

respect to P, also lies on S.

Ellipsoid

Reflecting property of ellipsoid and whispering galleries

Elliptic hyperboloid in one sheet The surface given by one of the equations 𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2

𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2

𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2 = 1 , is called an elliptic hyperboloid in one sheet.

• It is symmetric with respect to all coordinate planes.
• For the surface 𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2

1. The 𝑥- and 𝑦- intercepts are ±𝑎 and ±𝑏, there is no 𝑧- intercept.
2. The 𝑦𝑧- and 𝑧𝑥- traces are 𝑦^2 𝑏^2

𝑧^2 𝑐^2

and 𝑥^2 𝑎^2

𝑧^2 𝑐^2 = 1 , which are hyperbolas.

3. The 𝑥𝑦- trace is 𝑥^2 𝑎^2

𝑦^2 𝑏^2 = 1 , which is ellipse.

Elliptic hyperboloid of two sheets The surface given by any of the equations 𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2

𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2 = 1 and − 𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2 = 1 is called an elliptic hyperboloid of two sheets.

• It has symmetry with respect to all coordinate planes.
• The 𝑦-intercepts are ±𝑏 but there are no 𝑥-, 𝑧- intercepts. The surface do not cut the 𝑥- and 𝑧- axes.
• The 𝑥𝑦- and 𝑦𝑧- traces are 𝑦^2 𝑏^2

𝑥^2 𝑎^2

and 𝑦^2 𝑏^2

𝑧^2 𝑐^2 = 1 , which are hyperbolas.

• There is no trace on the 𝑧𝑥- plane.

Elliptic paraboloid The surface given by any one of the 𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑥^2 𝑎^2

𝑧^2 𝑐^2 = 𝑏𝑦 and 𝑦^2 𝑏^2

𝑧^2 𝑐^2

is called an elliptic paraboloid.

• For the surface 𝑥^2 𝑎^2

𝑦^2 𝑏^2

1. There is symmetry with respect to 𝑦𝑧- and 𝑧𝑥- plane.
2. If 𝑐 > 0 , the 𝑧 > 0 and if 𝑐 < 0 , then 𝑧 < 0. So the surface lies entirely above or below the 𝑥𝑦- plane according to the value of 𝑐.
3. The only intercept on any axis is 0; that is ( 0 , 0 , 0 ) is the only point of any axis on the surface.
4. Here, 𝑥𝑦- trace is the origin and the other two traces are the hyperbolas.

Hyperbolic paraboloid The surface given by any one of 𝑥^2 𝑎^2

𝑦^2 𝑏^2

𝑦^2 𝑏^2

𝑧^2 𝑐^2 = 𝑎𝑥 or 𝑥^2 𝑎^2

𝑧^2 𝑐^2 = 𝑏𝑦 is called a hyperbolic paraboloid. Consider the surface given by 𝑥^2 𝑎^2

𝑦^2 𝑏^2

• The surface is symmetric with respect to the 𝑦𝑧- and 𝑧𝑥- planes.
• The only point of the surface on any of the coordinate axes is ( 0 , 0 , 0 ).
• The 𝑥𝑦- trace is 𝑥 𝑎

𝑦 𝑏 , a pair of lines through origin.

• The 𝑦𝑧- and 𝑧𝑥- traces are parabolas.
• The section 𝑧 = 𝑧 1 is 𝑥^2 𝑎^2

𝑦^2 𝑏^2

Hyperbolic paraboloid