cancel
Showing results for
Did you mean:
cancel
Showing results for
Did you mean:

## Equations for Curves (and more)  4-Participant

# 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))

## 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

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

## 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

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

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 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.

19 REPLIES 19  22-Sapphire I

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.  4-Participant

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.  22-Sapphire I

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?   10-Marble

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  4-Participant

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.  1-Newbie

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  4-Participant

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.  1-Newbie

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.  1-Newbie

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  1-Newbie

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))  1-Newbie

thanks lot finally i got about the equation for curves  10-Marble

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  23-Emerald II

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

"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."  9-Granite

Not really sure what's "Copy - evaluation" command, but try searching for a term "Composite curve".  10-Marble

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  10-Marble

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  23-Emerald II
 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.  1-Newbie

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  1-Newbie

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 