Hyperbolic Paraboloid and Isometry

In summary, the hyperbolic paraboloid z=(x/a)^2 - (y/b)^2 can be rotated by an angle of π/4 in the +z direction to result in the surface z=(1/2)(x^2 + y^2) ((1/a^2)-((1/b^2)) + xy((1/a^2)-((1/b^2)). If a= b, this simplifies to z=2/(a^2) (xy). The quadric form of z=x^2 - y^2 can be written as v'Av + bv + c = 0 where A= [ 1 0 0; 0 -1 0; 0
  • #1
BrainHurts
102
0
If the hyperbolic paraboloid z=(x/a)^2 - (y/b)^2

is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface

z=(1/2)(x^2 + y^2) ((1/a^2)-((1/b^2)) + xy((1/a^2)-((1/b^2))

and if a= b then this simplifies to

z=2/(a^2) (xy)

suppose z= x^2 - y^2

does this mean that z=2xy ?

if so can someone tell me how to put z=x^2 - y^2 into it's quadric form? Also the rotation by the angle of π/4 is that just the typical rotation matrix Rz?
 
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  • #2
Hi BrainHurts! :smile:

(try using the X2 button just above the Reply box :wink:)
BrainHurts said:
suppose z= x^2 - y^2

does this mean that z=2xy ?

yes, x2 + y2 = 2* (x - y)/√2 * (x + y)/√2 = 2(Rx)(Ry) :wink:
if so can someone tell me how to put z=x^2 - y^2 into it's quadric form?

not following you :confused:
Also the rotation by the angle of π/4 is that just the typical rotation matrix Rz?

yes
 
  • #3
i actually got it,

a quadric is an equation that can be written in the form:

v'Av + bv + c = 0

so z=x^2 - y^2 by the above equation:

let v' is the transpose of v and v = [x y z]

A= [ 1 0 0; 0 -1 0; 0 0 0] and b = [0 0 -1]

so rotating v by the rotation matrix where the angle is pi/4 about the z axis gave me the results I was looking for. Thank you very much

sorry i don't see the button over the reply box but i'll try to do it next time!
 

1. What is a hyperbolic paraboloid?

A hyperbolic paraboloid is a three-dimensional surface that resembles a saddle. It is created by intersecting two sets of parallel lines at a constant angle, resulting in a curved surface with both convex and concave regions. It is also known as a saddle shape or a hypar.

2. How is a hyperbolic paraboloid formed?

A hyperbolic paraboloid is formed by using a mathematical equation to plot the points of intersection between two sets of parallel lines. The equation is x²/a² - y²/b² = z, where a and b are the distances between the parallel lines in the x and y directions, respectively.

3. What is isometry in relation to hyperbolic paraboloids?

Isometry is a type of transformation that preserves distance and angles between points. In relation to hyperbolic paraboloids, an isometry is a transformation that preserves the shape and curvature of the surface. This means that the resulting surface after an isometry is applied will be identical to the original surface.

4. What are some real-life applications of hyperbolic paraboloids?

Hyperbolic paraboloids have many practical applications in architecture and engineering. They can be used to create strong and stable structures, such as ramps, roofs, and bridges. They are also commonly used in the design of water slides and skate parks, as their saddle shape allows for smooth transitions between surfaces.

5. How are hyperbolic paraboloids used in mathematics and science?

Hyperbolic paraboloids are used in various fields of mathematics and science, such as differential geometry, calculus, and physics. They are also used in computer graphics to create 3D models and in computer-aided design (CAD) software for architectural and engineering purposes. They have also been studied extensively in topology and have applications in the study of minimal surfaces.

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