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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.

### License

GPL version 3.0.

### Version History

- version 1.0: released.