# A Practical Introduction to Computer Graphics by Ian O. Angell (auth.)

By Ian O. Angell (auth.)

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**Example text**

Clipping and Covering 37 Again space is divided into nine sectors by extending the edges of the cover. 1). Covering a line segment that joins the points (XI, YI) to (X2, Y2) is achieved by CALLCOVER(XI, YI, X2, Y2) where the covering sub routine is as folIows. SUBRßUT INE CßVER (Xl. Y 1 • X2. Y2j C RßUTINE Tß FIND THE CßvER Pß I1HS (XlD. YlDj AND (X2D. Y2j RESPECTIVELY. DY C CALCULATE THE MODES ßF (Xl. YlJ AND (X2. IY2j IF

3. Ifboth of the points defining the line segment lie on the same side of one pair of rectangle edges (that is, IXh IX2 = I or IYI *IY2 = 1) then the line lies completely outside the rectangle and must be drawn in total, for example, AB. When tbis is not the case we calculate (when necessary) the points (XID, YID) corresponding to 38 A Practicallntroduction to Computer Graphics (Xl, YI) and (X2D, Y2D) corresponding to (X2, Y2). If (Xl, YI) lies inside the rectangle, then (XlD, YlD) = (Xl, YI); if it is outside, then (XlD, YlD) is produced in the same way as in the elipping routine.

The matrices used are 3 x 3 square, even though the space is only two-dimensional! This is because the translation of the origin requires a combination of x and y-values together with a linear distance which is independent of these values. 1 'P"" ~ «p«son'«! by. oolumn "dm (~). lotion T = G oo 1/";2 ( rotation R = - ~/";2 sorueS= 1 1/";2 1/";2 O~) o Goo By combining the matrices S x R x T (x denotes matrix multiplication) we achieve the required change of system (note that the correct order of multiplication in the example is S x R x T and not T x R x S because the matrices 27 Transformations of 2D Space; Matrix Representation premuItiply the column vector representing a point) SXRXT=(g =G 0 I 0 0 1 0 ( V2 -1~V2 0)~ (-lbV2 0)~ (-1~V2 x W2 I/V2 x V2 1/V2 0 1/V2 1/V2 ~H~ 1/V2 1/V2 -1/V2 ) 1/V2 1 0 0 0 1 0 -~ ) -V2) 1/f2 Thus becomes V2 ( -1/V2 o and V2 1/V2 o (D becomes V2 1/V2 o which is consistent with the previous calculation.