Sketch equation curve

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, to model a parabola use either:

• Parametric:
  • x(t) = t
  • y(t) = t^2

or

• 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.

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

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

Units, Parameters, and Functions in equation curves

Balance the units in equations. Balancing units in equations 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 is red and an error glyph displays next to the mini-toolbar.

Equation curves support parameters and functions. 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

θ(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 Formats for equation curves

These are examples of formatting that certain operators and functions require.

Addition/Subtraction

Cartesian

x(t): 1 mm * t + 1

y(t): 1 mm * t - 1

Polar

r(t): 1 mm * t + 1

θ(t): 1 rad * t - 1 rad

Explicit Cartesian

y(x): x + 1

Explicit Polar

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

Miltiplication/Division

Cartesian

x(t): 2 mm * t

y(t): 2 mm / t

Polar

r(t): 2 mm * t

θ(t): 2 rad / t

Explicit Cartesian

y(x): 3 * x / 2

Explicit Polar

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

Exponents

Cartesian

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

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

Polar

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

θ(t): 1 rad * pow(t;2)

Explicit Cartesian

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

Explicit Polar

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

Trig Functions

Cartesian

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

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

Polar

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

θ(t): 1 rad * tan(1 rad * t)

Explicit Cartesian

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

Explicit Polar

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

Inverse Trig Functions

Cartesian

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

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

Polar

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

θ(t): acos(t)

Explicit Cartesian

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

Explicit Polar

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

Hyperbolic

Cartesian

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

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

Polar

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

θ(t): 1 rad * sinh(1 rad * t)

Explicit Cartesian

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

Explicit Polar

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

Log

Cartesian

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

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

Polar

r(t): 1 mm * log(t

θ(t): 1 rad * ln(t

Explicit Cartesian

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

Explicit Polar

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