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## Quaternions

Quaternions are an extension to the number system: real numbers may be thought of as points on a line, complex numbers as points in a plane, and quaternion numbers as points in four-dimensional space.

Apart from mathematical curiosity, quaternions are useful for specifying rotations in three dimensional space (e.g. in this game), and also for image analysis.

An introduction to quaternions and this scheme library is available here.

### Exported Procedures

[procedure] (make-rectangular w [x [y z]])

Constructs a number, complex number, or quaternion number depending on if given 1, 2 or 4 arguments.

```> (make-rectangular 1)
1
> (make-rectangular 1 2)
1+2i
> (make-rectangular 1 2 3 4)
1+2i+3j+4k```
[procedure] (make-polar mu theta [phi psi])

An alternative way of defining a quaternion, through the magnitude mu, the angle theta, the colatitude phi and longitude psi.

[procedure] (real-part n)

Returns the real part of a number.

```> (real-part (make-rectangular 1 2 3 4))
1```
[procedure] (imag-part n)

Returns the imaginary part (the i) of a complex or quaternion number.

```> (imag-part (make-rectangular 1 2 3 4))
2```
[procedure] (jmag-part n)

Returns the j part of a quaternion number.

```> (jmag-part (make-rectangular 1 2 3 4))
3```
[procedure] (kmag-part n)

Returns the k part of a quaternion number.

```> (kmag-part (make-rectangular 1 2 3 4))
4```
[procedure] (magnitude n)

The length of n treated as a vector.

[procedure] (angle n)

See make-polar.

[procedure] (number? n)

Checks if given argument is any kind of number.

```> (number? 3)
#t
> (number? 1+2i)
#t
> (number? (make-rectangular 1 2 3 4))
#t```
[procedure] (quaternion? n)

The same as number?.

[procedure] (= q1 ...)

Tests for equality of given numbers.

[procedure] (+ q1 ...)

Returns the sum of given numbers.

```> (+ 3 2+3i (make-rectangular 1 2 3 4))
6+5i+3j+4k```
[procedure] (- q1 ...)

Returns the difference of given numbers, associating to left.

[procedure] (* q1 ...)

Returns the product of given numbers.

```> (* 2 (make-rectangular 1 2 3 4))
2+4i+6j+8k```
[procedure] (/ q1 ...)

Returns quotient of given numbers, associating to left.

[procedure] (exp n)

Usual exp function extended to quaternions.

[procedure] (log n)

Usual log function extended to quaternions.

[procedure] (expt m n)

Usual expt function extended to quaternions.

[procedure] (sqrt n)

Usual sqrt function extended to quaternions.

[procedure] (sin n)

Usual sin function extended to quaternions.

[procedure] (cos n)

Usual cos function extended to quaternions.

[procedure] (tan n)

Usual tan function extended to quaternions.

[procedure] (asin n)

Usual asin function extended to quaternions.

[procedure] (acos n)

Usual acos function extended to quaternions.

[procedure] (atan n)

Usual atan function extended to quaternions.

[procedure] (atan y x)

Usual atan function extended to quaternions.

[procedure] (vector-part q)

Returns the quaternion with its real part set to 0.

[procedure] (colatitude q)

See make-polar.

[procedure] (longitude q)

See make-polar.

[procedure] (conjugate q)

Returns quaternion with same real part and sign of i j and k part of q reversed.

```> (conjugate (make-rectangular 1 2 3 4))
1-2i-3j-4k```
[procedure] (unit-vector q)

Returns a quaternion in same direction as q but of unit length; the real part of q is set to 0.

[procedure] (dot-product q1 q2)

Real parts of q1 and q2 must be 0. The dot product is the negative of the real part of q1 x q2.

[procedure] (cross-product q1 q2)

Real parts of q1 and q2 must be 0. The cross product is the vector part of q1 x q2.

### Author

The original version of this library was created by Dorai Sitaram.

This port to chicken scheme is by Peter Lane.