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Feb 02, 2012
11:11 AM

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Feb 02, 2012
11:11 AM

Equations for Curves (and more)

**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.

*The z variable is not necessary, but when used will give the curve that extra dimension. If in doubt, try z = t*10.*

Cartesian coordinates

x = 50 * t

y = 10 * sin (t * 360)

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)

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)

Cartesian coordinates

z = 0

x = 10 * t

y = log (10 * t +0.0001)

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)

Cartesian coordinate

a = 5

x = a * (cos (t * 360)) ^ 3

y = a * (sin (t * 360)) ^ 3

Cartesian coordinates

a = 10

x = 3 * a * t / (1 + (t ^ 3))

y = 3 * a * (t ^ 2) / (1 + (t ^ 3))

Cartesian coordinates

x = 4 * cos (t * (5 * 360))

y = 4 * sin (t * (5 * 360))

z = 10 * t

Cartesian coordinates

x = (4 * t)

y = (3 * t) + (5 * t ^ 2)

z = 0

Cartesian coordinates

X = 4 * cos (t * 3 * 360)

y = 2 * sin (t * 3 * 360)

z = 5

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

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)

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

/* "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

/* "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

/* "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)

/* "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 = t

theta = 10 + t * (20 * 360)

z = t * 3

Cylindrical coordinates

theta = t * 360

r = 10 +10 * sin (6 * theta)

z = 2 * sin (6 * theta)

Cylindrical coordinates

r = 5

theta = t * 3600

z = (sin (3.5 * theta-90)) +24 * t

Cylindrical coordinates

r = 5 + 0.3 * sin (t * 180) + t

theta = t * 360 * 30

z = t * 5

Cylindrical coordinates

R = 50 + t * (120)

Theta = t * 360 * 5

Z = 0

Cylindrical coordinates

a = 10

r = a * (1 + cos (theta))

theta = t * 360

Spherical coordinates

rho = 8 * t

theta = 360 * t * 4

phi = -360 * t * 8

Spherical coordinates

rho = 4

theta = t * 180

phi = t * 360 * 20

Spherical coordinates

rho = 20 * t ^ 2

theta = 60 * log (30) * t

phi = 7200 * t

Spherical coordinates

rho = 200 * t

theta = 900 * t

phi = t * 90 * 10

*Peruse the links for more equations and explanations as to how they work.*

- 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.

- 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

*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

* *

19 REPLIES 19

Feb 02, 2012
02:33 PM

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Feb 02, 2012
02:33 PM

Nice find! thanks for posting it.

Note: I suggest downloading by using the "View as PDF" button in the "Actions" menu in the upper right of this page. That way you can cut & paste and try some out. Interesting stuff here. Not a whole lot of practical applications in most cases (except involutes, etc.), but very cool for fun.

Feb 02, 2012
02:41 PM

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Feb 02, 2012
02:41 PM

Thank you, the models that you shared with me lead me to create this document. I haven't yet figured out how to recreate your models, but I have rediscovered my interest in the curve from equations command. However, I am aware that you used a different method to create your geometry. Maybe (or, hopefully) you'll share it with us later.

Feb 02, 2012
02:52 PM

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Feb 02, 2012
02:52 PM

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?

Feb 02, 2012
08:19 PM

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Feb 02, 2012
08:19 PM

Thanks for the post Kevin. I actually have a couple to add to your list. I discovered one really cool equation while trying to help someone else on Planet PTC> I haven't gotten a chance to write it up yet so I haven't posted it. When I do I'll send it to you, too so you can include it in your document (if you'd like).

Thanks!

-Brian

Feb 03, 2012
08:09 AM

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Feb 03, 2012
08:09 AM

I absolutely want them and will add them to the document. That goes for everyone. In time, I think I will also add images for each curve so a user, without having to try all of them, could decide which one they want to experiment with.

May 03, 2012
08:41 PM

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May 03, 2012
08:41 PM

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)

I have a question about the Double Arc Epicycloid. I'm looking to make a 4 arc version. Based on your equation, the (3 * t * 360) would change to (5 * t * 360). What I'm unclear about is the rest of the formula. The basic formula to create a epicycloid is:

x = (R + r) cos theta - r cos ((k + 1) theta)

y = (R+ r) sin theta - r sin ((k + 1) theta)

Where R = radius of the circle being revolved around, r = radius of the circle revolving around the R circle and k = the number of arcs.

What does "l" and "b" stand for in your equation? It looks like "l" stands for the "r", but I'm not sure about the "b". Also, what is the 3 shown before the "b" supposed to stand for?

The outside diameter at the tangent points of the arcs for my project is .7874 (or 2 cm).

Also, do you have a formula for creating a hypocycloid?

Thanks.

brian

May 03, 2012
10:00 PM

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May 03, 2012
10:00 PM

These equations aren't my own, but I can shed some light on them with the caveat that I don't have access to Creo and the equations anymore--If I did I could answer your questions better.

Anyway, the l & b from the equation are just multipliers, if I remember correctly. In order to maintain the desired shape of the curve, I think the intention was to only change the l & b values, thereby modifying the size of the curve, yet still being able to call it a double arc epicycloid. Because if you change the other values then the shape no longer deserves that name (I think). Did I make any sense?

Brian or Frank, from the posts above, may be able to help more.

Thanks for replying. It's a fun feature to play around with.

May 04, 2012
02:36 PM

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May 04, 2012
02:36 PM

First, it would help if I posted the equation correctly.

x = r (k+1) cos theta - r cos ((k+1) theta)

y = r (k+1) sin theta - r sin ((k+1) theta)

Anyways, I figured out that both "l" and "b" are both the same representation of "r". Changing them to the value I needed and changing each of the 3's to 5's (which is the k+1 value) made it work. Thanks for the input.

Dec 23, 2012
03:54 PM

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Dec 23, 2012
03:54 PM

hello!

I would deeply appreciate your help, as I am new to inserting equation into Pro E.

What is the cartesian or cylindrical equation for Logarithmic spiral that I should enter to the ptogram?!?!?

below the equation

http://mathdl.maa.org/images/upload_library/23/picado/seashells/espiraleng.html

Feb 17, 2014
11:34 PM

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Feb 17, 2014
11:34 PM

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))

Apr 11, 2015
06:01 AM

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Apr 11, 2015
06:01 AM

thanks lot finally i got about the equation for curves

Jun 07, 2016
10:50 AM

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Jun 07, 2016
10:50 AM

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

Jun 13, 2016
12:17 PM

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Jun 13, 2016
12:17 PM

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."**

Jun 13, 2016
02:09 PM

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Jun 13, 2016
02:09 PM

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

Jun 15, 2016
04:07 AM

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Jun 15, 2016
04:07 AM

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

Jun 15, 2016
04:09 AM

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Jun 15, 2016
04:09 AM

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

Jun 15, 2016
08:17 AM

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Jun 15, 2016
08:17 AM

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.

Mar 09, 2018
11:33 AM

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Mar 09, 2018
11:33 AM

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

Mar 09, 2018
11:37 AM

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Mar 09, 2018
11:37 AM

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