#### Examples.

- Input:
  trigsimplify(curvature([5*cos(t),5*sin(t)],t))
  Output:

  1
  5

- Input:
  curvature([2*cos(t),3*sin(t)],t)
  Output:

  ⎛
  ⎝

  6 cos2t+6 sin2t

  ⎞
  ⎠

  √

  9 cos2t+4 sin2t

  ⎛ ⎝ | 9 cos2t+4 sin2t | ⎞ ⎠ | 2

- Input:
  curvature([2*cos(t),3*sin(t)],t,pi/2)
  Output:

  3
  4

- Input:
  curvature(plot(x^2),point(1,1))
  (see Section 7.5 and Section 13.6.2.) Output:

  2
  25

  √

  5

The osculating_circle command finds and draws the osculating circle of a curve.

To find the osculating circle from a parameterization:

- osculating_circle takes three arguments:
  - C, a curve.
  - t, the parameter of the curve.
  - t0, a value of the parameter.
- osculating_circle(C,t,t0) draws and returns the osculating circle of the curve at the point specified by t0.

To find the osculating circle from a curve object:

- osculating_circle takes two arguments:
  - C, a curve.
  - p, a point on the curve.
- osculating_circle(C,p) draws and returns the osculating circle of C at the point p.

#### Examples.

- Input:
  osculating_circle(plot(x^2),point(1,1))
  Output:
- Input:
  equation(osculating_circle(plot(x^2),point(1,1)))
  Output:

  ⎛
  ⎝

  x+4

  ⎞
  ⎠

  2+

  ⎛
  ⎜
  ⎜
  ⎝

  y−

  7
  2

  ⎞
  ⎟
  ⎟
  ⎠

  2

  =

  125
  4

- Input:
  equation(osculating_circle([t^2,t^3],t,1))
  Output:

  ⎛
  ⎜
  ⎜
  ⎝

  x+

  11
  2

  ⎞
  ⎟
  ⎟
  ⎠

  2

  +

  ⎛
  ⎜
  ⎜
  ⎝

  y−

  16
  3

  ⎞
  ⎟
  ⎟
  ⎠

  2

  =

  2197
  36

The evolute command finds and draws the evolute of a curve.

To find the evolute from a parameterization:

- evolute takes two arguments:
  - C, a curve.
  - t, the parameter of the curve.
- evolute(C,t) draws and returns the evolute of the curve.

To find the evolute from a curve object:

- evolute takes one argument:
  C, a curve.
- evolute(C) draws and returns the evolute of C.

#### Examples.

- Input:
  evolute(plot(x^2))
  Output:
- Input:
  equation(evolute(plot(x^2)))
  Output:

  27 x2−16 y3+24 y2−12 y+2=0

- Input:
  equation(evolute([t^2,t],t))
  Output:

  16 x3−24 x2+12 x−27 y2−2=0