Juan Carlos Ponce Campuzano
Independent Mathematics Educator
\mathbf r
x=x\left(u,v\right)
y=y\left(u,v\right)
z=z\left(u,v\right)
\left(x,y,z\right)
PARAMETRIZATION OF SURFACES
A function of two real variables \(z=f(x,y)\)
\(\mathbb R\)
a unique real number \(z.\)
\((x,y)\)
\(D\)
\(f\)
assigns to each point in its domain \(D\subset \mathbb R^2\)
\(D\subset\mathbb R^2\)
f(x,y) = x^2-y^2
Surface
Surface
f(x,y) = x^2+ y^2
Surface
f(x,y) = \sqrt{1-x^2-y^2}
Surface
f(x,y) = ???
Surface
Sphere
f(x,y) = ???
Surface
Torus
f(x,y) = ???
Surface
Helicoid
f(x,y) = ???
\left(x_1,y_1\right)
z_1=f\left(x_1,y_1\right)
z_2=f\left(x_1,y_1\right)
f(x,y) = ???
z_1 \neq z_2
It doesn't fit the definition of function!
Parametrization of surfaces
How can we defined these surfaces?
Sphere
Torus
Helicoid
Parametrization of surfaces
Parametrization of surfaces
Parametrization of surfaces
\mathbf r
x=x\left(u,v\right)
y=y\left(u,v\right)
z=z\left(u,v\right)
\left(x,y,z\right)
Parametrization of surfaces
Parametrization of surfaces
Sphere
Torus
Helicoid
x = \cos u \sin v
y = \sin u \sin v
z = \cos v
x = \cos u\left(\frac{1}{2}\cos v + 2\right)
y = \sin u \left(\frac{1}{2} \cos v + 2\right)
z = \frac{1}{2}\sin v
x = u \cos v
y = u \sin v
z = v
0\leq u\leq 2\pi
0\leq u\leq \pi
0\leq u\leq 2\pi
0\leq u\leq 2\pi
-2\leq u\leq 2
-\pi\leq u\leq \pi
GeoGebra
Math3D.org
Desmos
Patrons:
Edward Huff, Abei, pmbem, Sophia Wood, Adam Parrott, Doug Kuhlman, bleh, Miguel Díaz, Ruan Ramon, Maciej Lasota, Christopher-Alexander Hermanns, Aarón Reyes, Gabriela Sofia Marin Sánchez, Jerome Siegler, Yashar Shoraka, Jeff Butterworth, Shaun MacMillan, Scott Pedersen, Ihsan Karabulut, Elias Sanchez Angarano, Emanuel Silva
Thanks for
watching!
Thanks for
watching!
Patrons:
Edward Huff, Abei, pmbem, Sophia Wood, Adam Parrott, Doug Kuhlman, bleh, Miguel Díaz, Ruan Ramon, Maciej Lasota, Christopher-Alexander Hermanns, Aarón Reyes, Gabriela Sofia Marin Sánchez, Jerome Siegler, Yashar Shoraka, Jeff Butterworth, Shaun MacMillan, Scott Pedersen, Ihsan Karabulut, Elias Sanchez Angarano, Emanuel Silva
Parametric surfaces
By Juan Carlos Ponce Campuzano
