How to draw a hyperbolic paraboloid John Ganci
![How to draw a hyperbolic paraboloid John Ganci Adjunct Math Faculty Richland CC, Dallas How to draw a hyperbolic paraboloid John Ganci Adjunct Math Faculty Richland CC, Dallas](https://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-1.jpg)
How to draw a hyperbolic paraboloid John Ganci Adjunct Math Faculty Richland CC, Dallas TX jganci@dcccd. edu Al Lehnen Math Instructor Madison Area Technical College, Madison WI alehnen@matcmadison. edu
![The steps • • • Identify the axis Identify the parabolas Draw two hyperbolas The steps • • • Identify the axis Identify the parabolas Draw two hyperbolas](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-2.jpg)
The steps • • • Identify the axis Identify the parabolas Draw two hyperbolas Connect the hyperbolas
![Identify the axis • Write equation in the form – u, v, and w Identify the axis • Write equation in the form – u, v, and w](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-3.jpg)
Identify the axis • Write equation in the form – u, v, and w are x, y, and z • u = x, v = y, w = z • u = y, v = x, w = z • u = z, v = x, w = y u = x, v = z, w = y u = y, v = z, w = x u = z, v = y, w = x • The one of degree 1, u, is the axis • If the equation is, • Then u = x, v = y, w = z, a = 1, b = 1 axis is the x-axis
![Identify the parabolas • Two parabolas are used for the sketch • The remaining Identify the parabolas • Two parabolas are used for the sketch • The remaining](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-4.jpg)
Identify the parabolas • Two parabolas are used for the sketch • The remaining two variables in the equation, v and w, are used for the parabolas • the “upper parabola” • the “lower parabola” • For • the “upper parabola” is • the “lower parabola” is
![Draw the parabolas • The upper parabola is in the uv-plane • The lower Draw the parabolas • The upper parabola is in the uv-plane • The lower](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-5.jpg)
Draw the parabolas • The upper parabola is in the uv-plane • The lower parabola is in the uw-plane • For – The upper parabola is in the xy-plane – The lower parabola is in the xz-plane • Determine “reasonable” limits for the domain values for the two parabolas – Upper: x = y^2; limit y to [-2, 2] or [-1, 1] – Lower: x = -z^2; limit z to [-2, 2] or [-1, 1]
![Draw the parabolas (1/2) The upper parabola x = y^2 -2 ≤ y ≤ Draw the parabolas (1/2) The upper parabola x = y^2 -2 ≤ y ≤](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-6.jpg)
Draw the parabolas (1/2) The upper parabola x = y^2 -2 ≤ y ≤ 2 0≤x≤ 4 Note upper bound for x
![Draw the parabolas (2/2) The lower parabola x = -z^2 Note lower bound for Draw the parabolas (2/2) The lower parabola x = -z^2 Note lower bound for](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-7.jpg)
Draw the parabolas (2/2) The lower parabola x = -z^2 Note lower bound for x -2 ≤ z ≤ 2 -4 ≤ x ≤ 0
![Draw the hyperbolas • One hyperbola for each of the parabolas • Drawn in Draw the hyperbolas • One hyperbola for each of the parabolas • Drawn in](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-8.jpg)
Draw the hyperbolas • One hyperbola for each of the parabolas • Drawn in planes perpendicular to the axis • Upper hyperbola drawn with upper parabola – The plane is the upper bound for the u variable • For • this is the plane x = 4 – Vertices are on the upper parabola • Lower hyperbola drawn with lower parabola – The plane is the lower bound for the u variable • For • this is the plane x = – 4 – Vertices are on the lower parabola
![Draw the hyperbolas (1/2) The upper hyperbola 4 = y^2 – z^2 --- or Draw the hyperbolas (1/2) The upper hyperbola 4 = y^2 – z^2 --- or](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-9.jpg)
Draw the hyperbolas (1/2) The upper hyperbola 4 = y^2 – z^2 --- or --1 = y^2/4 – z^2/4 Plane: x = 4 -2 ≤ z ≤ 2 -2 sqrt(2) ≤ y ≤ -2 or 2 ≤ y ≤ 2 sqrt(2)
![Draw the hyperbolas (2/2) The lower hyperbola -4 = y^2 – z^2 --- or Draw the hyperbolas (2/2) The lower hyperbola -4 = y^2 – z^2 --- or](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-10.jpg)
Draw the hyperbolas (2/2) The lower hyperbola -4 = y^2 – z^2 --- or --1 = z^2/4 – y^2/4 Plane: x = -4 -2 ≤ y ≤ 2 -2 sqrt(2) ≤ z ≤ -2 or 2 ≤ z ≤ 2 sqrt(2)
![Connect the hyperbolas • Connect the upper hyperbola, upper ends, to the lower hyperbola, Connect the hyperbolas • Connect the upper hyperbola, upper ends, to the lower hyperbola,](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-11.jpg)
Connect the hyperbolas • Connect the upper hyperbola, upper ends, to the lower hyperbola, upper ends • Connect the upper hyperbola, lower ends, to the lower hyperbola, lower ends • If the two hyperbola arcs are appropriately matched (see the document “An Interesting Property of Hyperbolic Paraboloids”), then these line segments lie on the surface of the hyperbolic paraboloid.
![Connect the hyperbolas (1/2) The upper ends Connect the hyperbolas (1/2) The upper ends](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-12.jpg)
Connect the hyperbolas (1/2) The upper ends
![Connect the hyperbolas (2/2) The lower ends Connect the hyperbolas (2/2) The lower ends](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-13.jpg)
Connect the hyperbolas (2/2) The lower ends
![Graph with sketch lines Graph with sketch lines](http://slidetodoc.com/presentation_image/0c5a2fa6df4dadae65e3aebf807d1531/image-14.jpg)
Graph with sketch lines
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