I wish I had found a list like this a few years ago. I've searched the web and compiled the following list for your enjoyment. Many are probably considered basic with a few very cool, complex curves thrown in. The list is divided into the coordinate systems that you will have to choose when creating the datum curve.
To initiate the command that will allow you to try these, go to Insert>Model Datum>Curve>From Equation.
After the Equations section see the section title Links to find PlanetPTC discussions and videos that have demonstrated and, in some cases, explained the curve from equation command in more detail with ways to incorporate relations and parameters.
Attached is a Creo Elements/Pro 5.0 part file with all of the equations included.
In the comments, please share any equations or links that you know. Other suggestions are welcome, too.
EQUATIONS
Cartesian Coordinates: x, y, & z
The z variable is not necessary, but when used will give the curve that extra dimension. If in doubt, try z = t*10.
Sine
Cartesian coordinates
x = 50 * t
y = 10 * sin (t * 360)
Rhodonea
Cartesian coordinates
theta = t * 360 * 4
x = 25 + (10-6) * cos (theta) +10 * cos ((10/6-1) * theta)
y = 25 + (10-6) * sin (theta) - 6 * sin ((10/6-1) * theta)
Involute
Cartesian coordinates
r = 1
ang = 360 * t
s = 2 * pi * r * t
x0 = s * cos (ang)
y0 = s * sin (ang)
x = x0 + s * sin (ang)
y = y0-s * cos (ang)
Logarithmic
Cartesian coordinates
z = 0
x = 10 * t
y = log (10 * t +0.0001)
Double Arc Epicycloid
Cartesian coordinate
l = 2.5
b = 2.5
x = 3 * b * cos (t * 360) + l * cos (3 * t * 360)
Y = 3 * b * sin (t * 360) + l * sin (3 * t * 360)
Star Southbound
Cartesian coordinate
a = 5
x = a * (cos (t * 360)) ^ 3
y = a * (sin (t * 360)) ^ 3
Leaf
Cartesian coordinates
a = 10
x = 3 * a * t / (1 + (t ^ 3))
y = 3 * a * (t ^ 2) / (1 + (t ^ 3))
Helix
Cartesian coordinates
x = 4 * cos (t * (5 * 360))
y = 4 * sin (t * (5 * 360))
z = 10 * t
Parabolic
Cartesian coordinates
x = (4 * t)
y = (3 * t) + (5 * t ^ 2)
z = 0
Eliptical Helix
Cartesian coordinates
X = 4 * cos (t * 3 * 360)
y = 2 * sin (t * 3 * 360)
z = 5
Disc Spiral 1
Cartesian coordinates
/* Inner Diameter
d = 10
/* Pitch
p = 5
/* Revolutions
r = 5
/* Height; use 0 for a 2D curve
h = 0
x = ((d/2 + p * r * t) * cos ((r * t) * 360))
y = ((d / 2 + p * r * t) * sin ((r * t) * 360))
z = t * h
Butterfly
a=cos(t*360)
b=sin(t*360)
c=cos(4*t*360)
d=(sin((1/12)*t*360))^5
x=b*(exp(a)-2*c+d)
y=a*(exp(a)-2*c+d)
Fish
a = cos (t * 360)
b = sin (t * 360)
/* As "c" increases the fish gets fatter until it transforms into a figure 8.
c = 10
x = (C*a-20*((b)^2)/1.5)
y = c * a * b
Cappa
/* "c" is a scaling variable
c=20
/* Revolutions
r=1
/* Height
h=0
x=c*cos(t*r*360)*sin(t*r*360)
y=c*cos(t*r*360)
z=t*h
Star
/* "a" & "b" are scaling variables
a=2
b=2
/* If, r=2/3 ----> astroid
/* If, r=2 ----> ellipse; when a=b, its a circle
/* r cannot equal 1
r=2/3
x=a*(cos(t*360))^(2/r)
y=b*(sin(t*360))^(2/r)
z=0
Bicorn
/* "c" is a scaling variable.
c=5
a=cos(t*360)
b=sin(t*360)
x=c*a
y=c*(a^2)*(2+a)/(3+b^2)
Talbots
/* "c" is a scaling variable.
c=10
a=cos(t*360)
b=sin(t*360)
x=C*a*(1+exp(2)*(b^2))
y=C*b*(1+exp(2)*(b^2))
Cylindrical Coordinates: r, theta, & z
Spiral
Cylindrical coordinates
r = t
theta = 10 + t * (20 * 360)
z = t * 3
Circle Spiral Column
Cylindrical coordinates
theta = t * 360
r = 10 +10 * sin (6 * theta)
z = 2 * sin (6 * theta)
Helical Wave
Cylindrical coordinates
r = 5
theta = t * 3600
z = (sin (3.5 * theta-90)) +24 * t
Basket
Cylindrical coordinates
r = 5 + 0.3 * sin (t * 180) + t
theta = t * 360 * 30
z = t * 5
Disc Spiral 2
Cylindrical coordinates
R = 50 + t * (120)
Theta = t * 360 * 5
Z = 0
Apple
Cylindrical coordinates
a = 10
r = a * (1 + cos (theta))
theta = t * 360
Spherical Coordinates: rho, theta, & phi
Butterfly Ball
Spherical coordinates
rho = 8 * t
theta = 360 * t * 4
phi = -360 * t * 8
Spherical Helix
Spherical coordinates
rho = 4
theta = t * 180
phi = t * 360 * 20
UFO
Spherical coordinates
rho = 20 * t ^ 2
theta = 60 * log (30) * t
phi = 7200 * t
Unnamed
Spherical coordinates
rho = 200 * t
theta = 900 * t
phi = t * 90 * 10
LINKS
Peruse the links for more equations and explanations as to how they work.
Web Links
- Involute Gears
- Power Tools: Curves by Equation
- This gives details about using Pro/E dimension references in the equation to give it a parametric touch.
Links to curve-from-equation Discussions on PlanetPTC:
- Curve from Equation Sample for Newbies
- Capto
- How to Create a Curve from the Equation? Does Anybody Know?
- Datum Curve from Equation Driven by Parameters
- Constant Force Spring Model
- Involute Helical Gear Geometry
Links for related PlanetPTC content:
If you think curves from equations are cool, then the following is right up your alley. These delve into the use of variable section sweep and the trajpar variable.
- From E-learning with Vladimir Palffy
- Fun with Helical Sweeps and Trajpar
- Conch Shell in Creo Parametric
- How to Create Twisted Wires
I'm kind of waiting to see if someone from PTC can't tell me how some of them, in particular, the wavy-washer, 3rd_curve and the coiled_spiral were created. 
How did I get the trajectory in the latter to circle back on itself 4 times without failing? 
Greetings,
I've been trying to turn the following into a functional epicycloidal curve from equations but have hit a wall:
I've reduced it to the following after trying it in its original format but it still blows up at zero, rotates the epicycloid CCW 5° or so and generally isn't the correct curve. This request for help may be beyond the scope of this forum in which case I'd be willing to talk consulting terms. Thank you.
A = 90
B = 12
C = 11
D = 3
x = A*cos(t*360) - B*cos((t*360) + atan ((sin(1-C)*(t*360)) / (A/(D*C) - cos(1-C)*(t*360)))) - D*cos(C*(t*360))
y = -A*sin(t*360) + B*sin((t*360) + atan ((sin(1-C)*(t*360)) / (A/(D*C) - cos(1-C)*(t*360)))) + D*sin(C*(t*360))
