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KevinBradberry1-Visitor
1-Visitor
February 2, 2012
Question

Equations for Curves (and more)

  • February 2, 2012
  • 20 replies
  • 76009 views

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

  1. Involute Gears
  2. Power Tools: Curves by Equation
    1. 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:

  1. Curve from Equation Sample for Newbies
  2. Capto
  3. How to Create a Curve from the Equation? Does Anybody Know?
  4. Datum Curve from Equation Driven by Parameters
  5. Constant Force Spring Model
  6. 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.


  1. From E-learning with Vladimir Palffy
    1. http://communities.ptc.com/blogs/vpalffy/2011/02/09/user-defined-springs
    2. http://communities.ptc.com/blogs/vpalffy/2011/11/27/sweep-and-trajpar
  2. Fun with Helical Sweeps and Trajpar
  3. Conch Shell in Creo Parametric
  4. How to Create Twisted Wires

    20 replies

    Show previous replies
    K
    knagarjun1-Visitor
    1-Visitor
    April 11, 2015

    thanks lot finally i got about the equation for curves

    T
    tleati12-Amethyst
    12-Amethyst
    June 7, 2016

    Hello,

    do you know whether it is possible to split the range of t into multiple intervals?

    for example I would like to set:

    for 0<t<0.5

    x=0

    y=t

    for 0.5<t<1

    x=t

    y=t^2

    I would really need this because if I build these two pieces with two separate curves and then merge them with the "Copy - evaluation" command, any sketch projecting the unified curve will always see it as two separate pieces and this causes me problems in sketch references when the curves themselves change.

    thanks

    bye

    T
    23-Emerald IV
    TomU23-Emerald IV
    23-Emerald IV
    June 13, 2016

    Short answer, no.  You can create the IF statements for the different ranges but they do not get evaluated correctly.

    Long answer from PTC:

    "The relation for trajpar does not support ranges (changes of function on different pieces of the trajectory, implemented with 'if' statements). Suggested using multiple variable section sweep features, with family tables, or defining geometrical breaks in the trajectory."

    J
    James621-Visitor
    1-Visitor
    June 13, 2016

    Not really sure what's "Copy - evaluation" command, but try searching for a term "Composite curve".

    T
    tleati12-Amethyst
    12-Amethyst
    June 15, 2016

    Hello Tom,

    thanks for your tip. What do you exactly mean with "they do not get evaluated correctly"?.

    Also, how could I define geometrical breaks ? by trimming the curve?

    bye

    T
    tleati12-Amethyst
    12-Amethyst
    June 15, 2016

    Hello James,

    with Copy-Evaluation I was meaning the action of selecting the curve (which highlights in green), Ctrl+C, then Ctrl+V and the copy evaluation menu opens (where you can add more curves to the copied one).

    Bye

    T
    23-Emerald IV
    TomU23-Emerald IV
    23-Emerald IV
    June 15, 2016

    What do you exactly mean with "they do not get evaluated correctly"?

    Well, I'm not a developer but from what I can tell the curve generator dynamically adjusts how many locations it needs to evaluate 't' for based on the results of the previous evaluations.  It will progressively reduce the step size (down to some minimum) until it meets some type of fitting criteria.  I don't know the exact algorithm, but if the first section of the curve uses an equation that makes it flat, then the entire curve will end up flat, regardless of what the equation for the other sections might be.

    how could I define geometrical breaks ? by trimming the curve?

    Yes, exactly.

    E
    EHSAN1-Visitor
    1-Visitor
    March 9, 2018

    Hi

    Thank you so much for your web site

    I want to plot NACA Aifoil by this equation but I can not view the Cure in PTC Creo Parametric?!

    M=0.08
    a0=0.2969
    a1=0.126
    a2=0.3516
    a3=0.2843
    a4=0.1015
    T=0.12
    p=0.4
    x=t-((T/.2)*((a0*t^.5)-(a1*t)-(a2*t^2)+(a3*t^3)-(a4*t^4)))*(Sin(360*atan((M/p^2)*(2*p-2*t))))
    y=((M/p^2)*(2*p*t-t^2))+((T/.2)*((a0*t^.5)-(a1*t)-(a2*t^2)+(a3*t^3)-(a4*t^4)))*(Cos(360*atan((M/p^2)*(2*p-2*t))))
    z=0

    E
    EHSAN1-Visitor
    1-Visitor
    March 9, 2018

    M=0.08
    a0=0.2969
    a1=0.126
    a2=0.3516
    a3=0.2843
    a4=0.1015
    T=0.12
    p=0.4
    x=t-((T/.2)*((a0*t^.5)-(a1*t)-(a2*t^2)+(a3*t^3)-(a4*t^4)))*(Sin(atan((M/p^2)*(2*p-2*t))))
    y=((M/p^2)*(2*p*t-t^2))+((T/.2)*((a0*t^.5)-(a1*t)-(a2*t^2)+(a3*t^3)-(a4*t^4)))*(Cos(atan((M/p^2)*(2*p-2*t))))
    z=0

    R
    razmosis13-Aquamarine
    13-Aquamarine
    May 19, 2026

    Pretty fascinating stuff here!  How can I alter the disc_spiral_cylindrical equation to make either the OD or ID of the spiral complete on a constant diameter?

    KenFarley
    KenFarley21-Topaz II
    21-Topaz II
    May 22, 2026

    There are tricks with the MIN and MAX functions that could kind of define a curve like that, but the results are not good from what I’ve seen when I tried it. The problem is that a spline is being used to define the curve, using points generated by the equations. If you introduce a discontinuity into the equation by defining a region of it that is supposed to be a constant “R”, at the point where it transitions from constant “R” to the remainder of the curve, there will be a “kink”. So the resultant spline will have a weird bump in it as the spline “adjusts” from one part of the curve to the other.

    You can see what this means by defining the curve given in the equation and then making a separate standard sketch on top of it that uses an arc/circle that is centered on the coordinate system, coincident with the endpoint of the curve at the smaller “R”. The end of the equation driven curve is not tangent to the arc.

    If you’re trying to define geometry that is to be used for a trajectory sweep, like if you are modeling a spiral wire for a thermostat, or a torsion spring, etc. you could probably use the spiral curve and a separate arc and live with the discontinuity. If you want the curve to be tangent at the transition from spiral to continuous arc, you will need to use other methods, like evalGraph() where you explicitly define the transition region(s). This way you’ll get a smooth curve that is closer to “reality” of what a wire would do.

    If you’re trying to define a spiral toolpath or something like that, you’re better off sketching a series of arcs that are tangent to each other and step out from the inner radius to the outer. I use this kind of thing to mill large circular pockets: drill a start hole and then step down with a pseudo-spiral path to rough out the “core”, then do a final pass with a finisher to make all the walls nice.

     

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