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Basic Geometric Shapes
Plane
\begin{array}{l}
x=1.5 u - 1 v + 1\\
y= u + v + 0.5\\
z= u +v + 1\\
\end{array}
-1.5\leq u \leq 1.5
-1.5\leq v \leq 1.5
Cylinder
\begin{array}{l}
x=r\cos u\\
y=r\sin u\\
z=h\\
\end{array}
0\leq u \leq 2\pi
0\leq h \leq a,\,a\in \mathbb R
Sphere
\begin{array}{l}
x = r\cos u \sin v\\
y = r\sin u \sin v\\
z = r\cos v\\
\end{array}
0\leq u \leq 2\pi
0\leq v \leq \pi
Cone
\begin{array}{l}
x = a \, u \cos v\\
y = b \, u \sin v\\
z = -c \, u\\
\end{array}
-1\leq u \leq 1
-\pi\leq v \leq \pi
Paraboloid
\begin{array}{l}
x = a \sqrt{u/h} \cos v\\
y = b \sqrt{u/h} \sin v\\
z = u\\
\end{array}
0\leq u \leq 1
0\leq v \leq 2\pi
Helicoid
\begin{array}{l}
x = u \cos v\\
y = u \sin v\\
z = c\, v\\
\end{array}
-2\leq u \leq 2
0\leq v \leq 2\pi
Catenoid
\begin{array}{l}
x = c \cosh (v/c)\cos u\\
y = c \cosh (v/c)\sin u\\
z = v\\
\end{array}
-\pi\leq u \leq \pi
-2\leq v \leq 2
Torus
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x=\cos u \left(\frac{r}{2} \cos v + R\right)
y=\sin u \left(\frac{r}{2} \cos v + R\right)
z = \frac{r}{2}\sin v
r\gt 0 ,\, R\gt0
Nature
Seashell
\begin{array}{l}
x = 2\left[ \exp\left(\displaystyle \frac{u}{6\pi}\right)-1\right]\cos u \cos^2 \left(\displaystyle \frac{1}{2}v\right)\\\\
y = 2\left[1-\exp\left(\displaystyle \frac{u}{6\pi}\right)\right]\sin u \cos^2 \left(\displaystyle \frac{1}{2}v\right)\\\\
z = 1 - \exp\left(\displaystyle \frac{u}{a}\right) - \sin v + \exp\left( \displaystyle \frac{u}{6\pi}\right)\sin v \\
\end{array}
0\leq u \leq 20
0\leq v \leq \pi
Apple
0\leq u \leq 2\pi
-\pi\leq v \leq \pi
x=\cos u \left(4+3.8 \cos v\right)
y=\sin u \left(4+3.8 \cos v\right)
z = \left( \cos v +\sin v -1 \right) \left( 1+ \sin v \right)\ln\left(1 - \pi v /10\right) + 7.5 \sin v
Horn
0\leq u \leq 1
-\pi\leq v \leq \pi
x=(a + u \cos v) \sin(b \,\pi\, u)
y=(a + u \cos v) \cos(b\, \pi\, u)+ c\, u
z = u \sin v
a,b,c\in \mathbb R
Heart
0\leq u \leq 2\pi
0\leq v \leq \pi
x= \big[4 \sin u - \sin(3 u)\big] \sin v
y= 2 \cos v
z = 1.2 \left(4 \cos u - \cos(2 u) - \frac{\cos(3 u)}{2} \right) \sin v
Snails & Mussels
u_{\text{min}}\leq u \leq u_{\text{max}}
0\leq v \leq 2\pi
x= (h + a \cos v) \exp(w \, u) \cos(c \, u)
y= (h + a \cos v) \exp(w \, u) \sin(c \, u)
z = (k + b \sin v) \exp(w \, u)
0\lt a,b,c\in \mathbb R; h,k,w\in \mathbb R
Egg
0\leq u \leq a
0\leq v \leq 2 \pi
x= c \sqrt{u (u - a) (u - b)} \sin v
y= u
z = c \sqrt{u (u - a) (u - b)} \cos v
0\lt a,b,c\in \mathbb R; a\lt b
Rose
s = \sin \left(\dfrac{\pi}{2} \exp\left(-\dfrac{v}{8\pi}\right)\right)
\alpha = 1 - \dfrac{1}{2}\left[\frac{5}{4}\left( 1-\dfrac{\text{Mod}\left(3.6 v, 2\pi \right) }{\pi}\right)^2 - \dfrac{1}{4}\right]^2
\beta = 1.95653 u^2 \left(1.27689 u -1\right)^2
r = \left(u + \beta\, c\right)s
x = r \,\alpha\,\sin v
y = r \,\alpha \,\cos v
z = u \, c - \beta s^2
0\leq u \leq 1
0\leq v \leq 17 \pi
c = \cos \left(\dfrac{\pi}{2} \exp\left(-\dfrac{v}{8\pi}\right)\right)
Objects
Goblet
\begin{array}{l}
x = \cos u \cos (2v)\\
y = \sin u \cos( 2v)\\
z = -\sin v \\
\end{array}
0\leq u \leq 2\pi
0\leq v \leq \pi
Bowtie
\begin{array}{l}
x = \dfrac{\sin u}{\sqrt{2}+\sin v}\\\\
y = \dfrac{\sin u}{\sqrt{2}+\cos v}\\\\
z = \dfrac{\cos u}{1+\sqrt{2}}\\
\end{array}
-\pi\leq u \leq \pi
-\pi\leq v \leq \pi
Pillow
\begin{array}{l}
x = \cos u \\
y = \cos v \\
z = a \sin u \sin v\\
\end{array}
0\leq u \leq \pi
-\pi\leq v \leq \pi
Trash can
\begin{array}{l}
x = (b+v)\cos u \\
y = v \sin u \\
z = a \, v^2\\
\end{array}
0\leq u \leq 2\pi
0\leq v \leq 2
Umbrella
0\leq u \leq 1
0\leq v \leq 2\pi
\begin{array}{l}
x = u^{1/3} \bigg[ (R-r)\cos v + r \cos ((n-1)v) \bigg] \\\\
y = u^{1/3} \bigg[ (R-r)\sin v - r \sin ((n-1)v) \bigg]\\\\
z = h (1-u)\\
\end{array}
r=\frac{R}{n}
Hyperbolic surfaces
Hyperbolic
0\leq u \leq 25
0\leq v \leq 1
x=\frac{\cos u}{u}
y=H v
z = \frac{\sin u }{u}
Spiral
Hyperbolic
-4\leq u \leq 4
-4\leq v \leq 4
x= \dfrac{\sinh v}{1+\cosh u \cosh v}\cos(a \,u)
y= \dfrac{\sinh v}{1+\cosh u \cosh v}\sin(a \,u)
z = \dfrac{\sinh(u)\cosh v}{1+\cosh u \cosh v}
Helicoid
Hyperbolic
-\frac{\pi}{2}\leq u \leq \frac{\pi}{2}
-\pi \leq v \leq \pi
x=(\cos u \cos v)^3
y=(\sin u \cos v)^3
z = \sin^3 v
Octahedron
Hyperbolic
-\frac{3}{2}\leq u \leq \frac{3}{2}
-\frac{\pi}{2} \leq v \leq \frac{\pi}{2}
x=\frac{\sinh(2u)}{ \cos(10 u) + \cosh(2u)}
y=v
z = \frac{\sin(10u)}{ \cos(10 u) + \cosh(2u)}
Tangent spiral
Hyperbolic
0\leq u \leq 1
0\leq v \leq 2\pi
x=u\cos v
y=u \sin v
z = u^2 \cos v \sin v
Paraboloid
Surfaces named
after mathematicians
Möbius band
-1\leq u \leq 1
0\leq v \leq 2\pi
x= \left[R +u \cos\left(\frac{v}{2}\right)\right]\cos v
y= \left[R +u \cos\left(\frac{v}{2}\right)\right]\sin v
z = u \sin \left(\frac{v}{2}\right)
R\gt0
Klein bottle
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x = \left\{\begin{array}{l}
\text{If\;\;} 0\leq u \lt \pi: \\
\;\; a \cos u \left(1+ \sin u\right) + r \cos u \cos v \\
\text{If\;\;} \pi\leq u \leq 2\pi: \\
\;\; a \cos u \left(1+ \sin u\right) + r \cos (v+\pi)
\end{array}
\right.
y = \left\{\begin{array}{l}
\text{If\;\;}0\leq u \lt \pi:\\
\;\;b \sin u + r \sin u \cos v\\
\text{If\;\;}\pi\leq u \leq 2\pi:\\
\;\;b \sin u
\end{array}
\right.
z = r \sin v
r = 3 \left(1- \frac{\cos u}{2}\right)
Klein bottle
0\leq u \leq 4\pi
0\leq v \leq 2\pi
x= \cos u \left[\cos(u / 2) \left(\sqrt{2} + \cos v \right) + \sin(u / 2) \sin v \cos v\right]
y=\sin u \left[\cos(u / 2) \left(\sqrt{2} + \cos v\right) + \sin(u / 2) \sin v \cos v\right]
z = - \sin(u / 2) \left[ \sqrt{2} + \cos v + \cos(u / 2) \sin v \cos v)\right]
Nordstrand
Lawson bottle
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x=\frac{\sin u \sin v - \sin (u/2) \cos v}{\sqrt{2}(1+w)}
y=\frac{\cos u \sin v}{1+w}
z = \frac{\cos(u/2)\cos v}{1+w}
w=\frac{\sin u \sin v + \sin (u/2) \cos v}{\sqrt{2}}
Dini's surface
0\leq u \leq 4\pi
0.01\leq v \leq 2
x=a \cos u \sin v
y=a \sin u \sin v
z = a\left[ \cos v + \ln\left(\tan\left(\frac{v}{2}\right) \right)\right]+ b\, u
Enneper's surface
-2\leq u \leq 2
-2\leq v \leq 2
x=u -\frac{u^3}{3} + uv^2
y= v - \frac{v^3}{3} + u^2v
z = u^2 - v^2
Dupin cyclide
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x=\frac{d ( c-a \cos u \cos v) + b^2 \cos u}{h}
y= \frac{b \sin u (a-d \cos v)}{h}
z = \frac{b \sin v (c \cos u -d)}{h}
h = a - c\cos u \cos v
Maeder's owl
0\leq u \leq 4\pi
0.01\leq v \leq 1
x=v\cos u -\frac{1}{2}v^2 \cos(2u)
y=- v\sin u -\frac{1}{2}v^2 \cos(2u)
z = \frac{4}{3} r^{3/2}\cos(3u/2)
Bernat Ancochea's surface
0\leq u \leq 1.2
0\leq v \leq 2\pi
x=u \cdot f_x + q\cdot g_x
y=u \cdot f_y + q\cdot g_y
z = u \cdot f_z + q\cdot g_z
f_x = 4.503 \cos v
f_y = 3.266 \sin v
f_z = 0
g_x = s \cdot 3.006 \cos v
g_x = s \cdot 2.266 \sin v
g_z = - s\cdot 3.006 \cos v
q=0.251u^{3}+0.389u^{2}-1.64u+1
Morin surface
0\leq u \leq \pi
0\leq v \leq \pi
x=K\bigg[\frac{2}{n-1} \cos u \cos((n-1)v) + \sqrt{2} \sin u \cos v\bigg]
y=K\bigg[\frac{2}{n-1} \cos u \sin((n-1)v) + \sqrt{2} \sin u \sin v\bigg]
z =K \cos u
K = \frac{\cos u}{\sqrt{2} - k \sin (2u)\sin (n\,v)}
Tori
8 Figure
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \cos u\big[c + \sin v \cos u - \sin (2v) \sin v / 2 \big]
y= \sin u \sin v + \cos u \sin (2v)/2
z =\sin u\big[c + \sin v \cos u - \sin (2v) \sin v / 2 \big]
0\lt c\in \mathbb R
Torus
Antisymmetric
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \big[R + r \cos v (a + \sin u)\big]\cos u
y= \big[R + r \cos v (a + \sin u)\big]\sin u
z =r \sin v\big[a + \sin u \big]
0\lt a\in \mathbb R
Torus
Braided
0\leq u \leq 8\pi
0\leq v \leq 2\pi
x= r\cos v \cos u + R \cos u \big[1+ a \cos ( n\,u)\big]
y= 2.5 \big[r \sin v + a \sin (n\, u) \big]
z =r\cos v \sin u + R \sin u \big[1+ a \cos ( n\,u)\big]
a\in \mathbb R
Torus
Twisted Eight
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \big[ R + r \left(\cos (u/2) \sin v - \sin(u/2) \sin (2v)\right) \big]\cos u
y= \big[ R + r \left(\cos (u/2) \sin v - \sin(u/2) \sin (2v)\right) \big]\sin u
z =r \big[\sin (u/2) \sin v + \cos (u/2 ) \sin 2v\big]
Torus
Twisted
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \big[4 + R \cos(t \, u + v)\big] \cos u
y= \big[4 + R \cos(t \, u + v)\big] \sin u
z = R \sin(t \, u + v);
Torus
R = \left(\cos^n v + \sin ^n v\right)^{-1/n}
2\lt n\in\mathbb Z^+ \text{\;even;}\; t\in \mathbb Z^+
Umbilic
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \sin u \big[7 + \cos ( u/3 - 2v) + 2 \cos (u/3+v)\big]
y= \cos u \big[7 + \cos ( u/3 - 2v) + 2 \cos (u/3+v)\big]
z = \sin (u/3 - 2v) + 2 \sin (u/3 + v)
Torus
Bianchi-Pinkall
0\leq u \leq 2\pi
0\leq v \leq \pi
x= \cos (u+v) \cos \gamma
y= \sin (u+v ) \cos \gamma
z = \cos (u-v ) \sin \gamma
Flat Tori
\gamma = a + b \sin (2n\, v)
w = \sin (u-v ) \sin \gamma
Sinusoidal surfaces
Sine
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \sin u
y= \sin v
z =\sin(u+v)
Cosine
0\leq u \leq \pi
0\leq v \leq 2\pi
x= \cos u
y= \cos v
z =\cos(u+v)
Sine Cube
0\leq u \leq \pi
0\leq v \leq 2\pi
x= \sin u \sin v
y= \cos u \sin v
z =\cos u \cos v
Sinusoidal Cone
-10\leq u \leq 10
-2\pi\leq v \leq 2\pi
x= u \cos v
y= u \sin v
z =k \, u \cos (n\,v)
k, n \in \mathbb R
Sine waves
-15\leq u \leq 15
-15\leq v \leq 15
x= u
y= a\sin\left(b \sqrt{u^2+v^2}\right)
z =v
a, b \in \mathbb R
Cosine waves
-15\leq u \leq 15
-15\leq v \leq 15
x= u
y= a\cos\left(b \sqrt{u^2+v^2}\right)
z =v
a, b \in \mathbb R
Spiral waves
0\leq u \leq 2\pi
0\leq v \leq 20
x=v \cos u
y= a \cos (b\,u + c\, v)
z = v \sin u
Wave ball
0\leq u \leq 14
0\leq v \leq 2 \pi
x= u \cos(\cos u) \cos v
y= u \cos(\cos u) \sin v
z =u \sin(\cos u)
Knots
Trefoil knot
0\leq u \leq 4\pi
0\leq v \leq 2\pi
x= \cos u \cos v + \frac{3}{2} \cos u \left(3+\sin\left(\frac{3}{2} u\right)\right)
y= \sin u \cos v + \frac{3}{2} \sin u \left(3+\sin\left(\frac{3}{2} u\right)\right)
z =\sin v + 2\cos\left(\frac{3}{2} u\right)
Figure-8 knot
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \frac{\sin(3u) \left( e \sin(4u) + 1 \right)}{\left(1.5 + \frac{\sin(1.5v)}{4} \right) (e \sin(4u) - 1)}
y = \frac{\cos(3u) \left( e \sin(4u) + 1 \right)}{\left(1.5 + \frac{\sin(1.5v)}{4} \right) (e \sin(4u) - 1)}
z =\frac{-2h \sin(2u) \left( e \sin(4u) + 1 \right)}{e \sin(4u) - 1} + 0.1 \cos(1.5v)
Torus knots
0\leq u \leq 2\pi
0\leq v \leq 2\pi
x= \left( R_1 + R_2 \cos(p\,u) + r \cos v \right) \cos(q \,u)
y= \left( R_1 + R_2 \cos(p\,u) + r \cos v \right) \sin(q u)
z = r \sin v + R_2 \sin(p\,u)
Tranguloid trefoil
-\pi\leq u \leq \pi
-\pi\leq v \leq \pi
x= \dfrac{2 \sin(3 u)}{2 + \cos v}
y= \frac{2 (\sin u + 2 \sin(2 u))}{2 + \cos(v + 2 \pi/3)}
z = \frac{1}{4} \big( \cos u - 2 \cos(2 u) \big) (2 + \cos v ) \big(2 + \cos(v + 2 \pi/3)\big)
References
This gallery would not exist without the incredible mathematical work contributed by many people around the world:
- Encyclopédie des formes mathématiques: Surfaces.
- Henry Segerman. (2016). Visualizaing Mathematics with 3D printing. Johns Hopkins University Press: USA.
- Jürgen Meier. (2020). Parametrische Flächen und Körper.
- Minimal Surfaces. Wolfram Research, Inc.
-
Miscellaneous Surfaces. Wolfram Research, Inc.
- Paul Bourke. Geometry, Surfaces, Curves, Polyhedra.
- Paul Nylander. www.bugman123.com
- Ruled Surfaces. Wolfram Research, Inc.
- S.N. Krivoshapko & V.N. Ivanov. (2015). Encyclopedia of Analytical Surfaces. Springer Verlag.
- Surfaces of Revolution. Wolfram Research, Inc.
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