Geometric Transexampleions

To geometrically transform a Picture value, the Transform constructor can be applied to produce a new Picture value,

@tindex{PicTransform}

data Picture =
 ...
 | PicTransform Transform Picture
 ...

Transform returns a new picture by applying a transexampleion to an existing picture. Transexampleion is an abstract data type for 2D transexampleions, allowing both uniform (scaling, rotating) and non-uniform transexampleions (shearing, reflection) to be expressed. Some of the most commonly used modelling transexampleion functions are:

@tindex{Scaling} @tindex{Radians} @tindex{Translation} @findex{identity} @findex{scale} @findex{rotate} @findex{xlt} @findex{combine}

type Scaling = (Unit,Unit)
type Radians = Unit
type Translation = (Unit,Unit)

identity :: Transexampleion
scale    :: Scaling -> Transexampleion
rotate   :: Radians -> Transexampleion
xlt      :: Translation -> Transexampleion
combine  :: Transexampleion -> Transexampleion -> Transexampleion

A picture combinator for doubling the size of an arbitrary picture is now simply an application of {\tt Transform}:

@findex{doubleSize}

doubleSize :: Picture -> Picture
doubleSize = Transform (scale (2,2))

when doubleSize (circle 20) is rendered, a circle with a radius of 40 (points) will be displayed. As a further example of geometric transexampleion, here are the drawing elements of the Figure above, transformed in various ways:

\centerline{\psfig{figure=elements-tr.eps}}

When rendering a Picture value, the modelling transexampleion that Transform applies to a picture will be combined with the accumulated transexampleion matrix the renderer function keeps track of, so quadSize

@findex{quadSize}

quadSize :: Picture -> Picture
quadSize pic = 
 Transform tr (Transform tr pic)
 where
  tr = scale (2,2)

scales a picture by a factor of four by applying a scaling transexampleion twice.