Surfaces from MathCurve

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2014-09-20 22:39

In collections surface 3D /3D-Oberflächen 1 surface 3D /3D-Oberflächen 1 surface 3D surface 3D

This example demonstrates the construction and visualization of two 3D surface types, the Astroid and the Berlingot, sourced from MathCurve. It involves core mathematical concepts such as parametric equations, polynomial functions, solid geometry, and surface properties. Key GeoGebra commands and objects used include conditional statements (If), parameters (a, b, c, Umin, Umax, Vmin, Vmax), predefined surface models (BohemianDome, Astroid, Berlingot, Romaine, CrossCap), and a parameter 'n'. Suitable for high school students and math enthusiasts studying 3D surfaces and their properties.

GeoGebra Commands & Parameters

If(n ≟ 1, Surface[((a * cos(v)^(3)) * cos(u)^(3)), ((b * cos(v)^(3)) * sin(u)^(3)), (c * sin(v)^(3)), u, Umin, Umax, v, Vmin, Vmax]) → astroid
If(n ≟ 2, Surface[((a * (1 + u)) * cos(v)), ((b * (1 - u)) * sin(v)), (c * u), u, Umin, Umax, v, Vmin, Vmax]) → berlingot
If(n ≟ 3, Surface[(a * cos(u)), (b * cos(v)), (a * sin(u)) + (b * sin(v)), u, Umin, Umax, v, Vmin, Vmax]) → bohemianDome
If(n ≟ 4, Surface[((a * cos(v)) * sin((2 * u))), ((b * sin(v)) * sin((2 * u))), (c * (sin(u)^(2) - (cos(u)^(2) * cos(v)^(2)))), u, Umin, Umax, v, Vmin, Vmax]) → crossCap
If(n ≟ 5, Surface[((a * sin((2 * u))) * cos(v)), ((b * sin((2 * u))) * sin(v)), ((c * sin(u)^(2)) * sin((2 * v))), u, Umin, Umax, v, Vmin, Vmax]) → romaine

AI Drawing Prompt Reference

Draw three 3D surfaces defined by parametric equations: the first with x=a*cos(u)^3*cos(v)^3, y=b*cos(u)^3*sin(v)^3, z=c*sin(v)^3; the second with x=a*(1+u)*cos(v), y=b*(1-u)*sin(v), z=c*u; and the third with x=a*cos(u), y=b*cos(v), z=a*sin(u)+b*sin(v), switching between them by adjusting the parameter n from 1 to 5 to display different geometric shapes.

The prompt above was compiled from the original drawing commands and is for reference only.