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# Sketch equation curves

Equation curves are used to model complex geometry such as gear tooth profiles or sweep paths for hydraulic pumps. To generate an equation curve, specify the equations to define the curve and a range to evaluate the equations.

The equations can be parametric, where X and Y vary as a function of a variable t, or explicit, where Y varies as a function of X.

For example, a parabola can be modeled using either parametric or explicit.

Parametric:

• x(t) = t
• y(t) = t^2

Explicit:

• y(x) = x^2 / 1 mm

2D equation curves support both Cartesian and Polar coordinate systems. The coordinate system is specified in the Equation Curves mini-toolbar.

Units, Parameters, and Functions

Units must be balanced in equations. This often requires multiplying or dividing by 1 or multiple units of length. When the units are not a single unit of length, the equation text will be red and an error glyph will display next to the mini-toolbar.

Parameters and functions are supported in equation curves. The exception for parameters is that you cannot have a parameter named "t" because that is used for the variable in equations. Unsupported functions are:

• Floor
• Ceiling
• Abs
• Sign
• % Modulo

Example Equations

 Parametric Cartesian x(t): 4 * cos(1 rad * t) / sqrt(t) * 1 mm y(t): 4 * sin(1 rad * t) / sqrt(t) * 1 mm tmin: 0.01 tmax: 6 * PI Explicit Cartesian y(x): x * sin(1 rad * x / 1 mm) xmin: -1 * PI xmax: 6 * PI Parametric Polar r(t): t * 1 mm q(t): cos(t * 1 rad) * 1 rad * 5 * PI / 4 tmin: -5 * PI tmax: 5 * PI Explicit Polar r(a): sqrt(a / 1 rad) amin: 0.01 amax: 12 rad * PI

Example Equation Format

This table shows examples of the formatting required to use certain operators and functions.

Cartesian Polar Explicit Cartesian Explicit Polar

x(t): 1 mm * t + 1

y(t): 1 mm * t - 1

r(t): 1 mm * t + 1

y(x): x + 1

r(a): 1 mm * a / 1 rad + 1

Multiplication/Division

x(t): 2 mm * t

y(t): 2 mm / t

r(t): 2 mm * t

y(x): 3 * x / 2

r(a): 3 mm * a / 2 rad

Exponents

x(t): (t^2) * 1 mm

y(t): 1 mm * pow(t;2)

r(t): 1 mm * (t^2)

y(x): 1 in * (x / 1 mm)^3

r(a): 1 mm * ((a / 1 rad)^3)

Trig Functions

x(t): 1 mm * sin(1 rad * t) + 1 mm * cos(1 rad * t)

y(t): 1 mm * tan(1 rad * t)

r(t): 1 mm * cos(1 rad * t) + 1 mm * sin(1 rad * t)

y(x): 1 mm * sin(1 rad * x / 1 mm)

r(a): 1 mm * cos(a)

Inverse Trig Functions

x(t): 1 mm * asin(t) / 1 rad + 1 mm * asin(t) / 1 rad

y(t): 1 mm * atan(t) / 1 rad

r(t): 1 mm * asin(t) / 1 rad

q(t): acos(t)

y(x): 1 mm * acos(x / 1 mm) / 1 rad

Hyperbolic

x(t): 1 mm * sinh(1 rad * t) + 1 mm * cosh(1 rad * t)

y(t): 1 mm * tanh(1 rad * t)

r(t): 1 mm * cosh(1 rad * t)

y(x): 1 mm * tanh(1 rad * x / 1 mm)

r(a): 1 mm * cosh(a)

Log

x(t): 1 mm * ln(t)

y(t): 1 mm * log(t)

r(t): 1 mm * log(t)

y(x): 1 mm * ln(x / 1 mm)

r(a): 1 mm * ln(a / 1 rad)

Procedures

Create equation curves

To create an equation curve, select a part face or work plane to use as the sketch plane . In an empty file, the sketch plane is preset, and you can start to sketch.

Show Me how to create and use equation curves

Create a Parametric Cartesian equation curve

Cartesian equation curves use X, Y coordinates. Parametric equation curves use equations to define x and y as a function of a variable t.

1. On the ribbon, click Sketch tab Draw panel Equation Curve.
2. Select Parametric.
3. Select Cartesian.
4. Enter the equation for x as a function of t in the x(t) field.
5. Enter the equation for y as a function of t in the y(t) field.
6. Enter the minimum and maximum values for t in the tmin and tmax fields.
7. Click OK to create the curve and exit the command. Click Apply to create the curve and remain in the command.

Create an Explicit Cartesian equation curve

Cartesian equation curves use X, Y coordinates. Explicit equation curves use a single equation to define y as a function of x.

1. On the ribbon, click Sketch tab Draw panel Equation Curve.
2. Select Explicit.
3. Select Cartesian.
4. Enter the equation for y as a function of x in the y(x) field.
5. Enter the minimum and maximum values for x in the xmin and xmax fields.
6. Click OK to create the curve and exit the command. Click Apply to create the curve and remain in the command.

Create a Parametric Polar equation curve

Polar equation curves specify coordinates as a distance (r) and an angle (q). Parametric equation curves use equations to define r and q as a function of a variable t.

1. On the ribbon, click Sketch tab Draw panel Equation Curve.
2. Select Parametric.
3. Select Polar.
4. Enter the equation for r as a function of t in the r(t) field.
5. Enter the equation for q as a function of t in the q(t) field.
6. Enter the minimum and maximum values for t in the tmin and tmax fields.
7. Click OK to create the curve and exit the command. Click Apply to create the curve and remain in the command.

Create a Explicit Polar equation curve

Polar equation curves specify coordinates as a distance (r) and an angle (a). Explicit equation curves us a single equation to define r as a function of a.

1. On the ribbon, click Sketch tab Draw panel Equation Curve.
2. Select Parametric.
3. Select Cartesian.
4. Enter the equation for r as a function of a in the r(a) field.
5. Enter the minimum and maximum values for a in the amin and amax fields.
6. Click OK to create the curve and exit the command. Click Apply to create the curve and remain in the command.

Edit equation curves

In an active sketch, use one of the following methods to edit an equation curve:

• Right-click the curve in the graphics window or the browser then select Edit Equation Curve to display the mini-toolbar. You can modify the equations, change the coordinate system, change the equation type (parametric or explicit) or edit the range.
• Drag the curve to reposition it or drag geometry constrained to the curve.
• Add constraints to control the relationship between the equation curve and other geometry.
• Add dimensions between the curve and other geometry to control the position of the curve.
• Right-click on the curve and select Display Curvature to analyze the curvature of the equation curve.

References

Equation curves

 Access: Ribbon: Sketch tab Draw panel Equation Curve
 x(t) or r(t) For parametric equations. Specify x or r as a function of t. For example x(t): t^2. y(t) or q(t) For parametric equations. Specify y or q as a function of t. For example, y(t): t^2. y(x) or r(a) For explicit equations. Specify an equation for y or r as a function or x or a. For example, y(x): x^2. tmin, xmin, or amin Specify the minimum value to evaluate. The t, x, or a will be displayed depending on the Parametric/Explicit and Cartesian/Polar settings. tmax, xmax, or amax specify the maximum value to evaluate. The t, x, or a will be displayed depending on the Parametric/Explicit and Polar/Cartesian settings. Parametric or Explicit Parametric equations use two equations to evaluate x and y or r and q. Explicit equations use one equation to evaluate y or r and you specify a range for x or a. Cartesian or Polar Determine whether to specify equations using a Cartesian (x,y) coordinate system or polar (r, q) coordinate system. Show units When not checked, the equation is displayed as it was entered by the user. When checked, the equation is displayed with all units parsed. Show transform Available when editing an equation curve. Displays the values for X, Y, and Rotation for the transform. These values will be zero until the curve has been moved by dragging, constraints, or dimensions.