1. Graphics Vector graphics 2D only. Overview of common functions and parameters.. Drawing in Matlab. We will see how a common file format stores the information. Looking at similar functions in PDF.

2. Graphics Source of confusion. Vectors vs Bitmaps. Here we are concerned with vector graphics.

3. Vector Graphics Is to pictures what MIDI is to sound. Uses lines, predefined shapes, curves and (predefined text). Can be very compact. Good for plotters. Converted to bitmap for monitor display.

4. Co-ordinate systems Vector graphics based upon an x, y co-ordinate system. The x co-ordinate runs from left to right across the screen. The y co-ordinate usually runs from the bottom (= 0) of the image to the top, but sometimes from top (= 0) to the bottom.

5. Co-ordinate systems To set up co-ordinate system in Matlab. haxes=axes –Sets up a co-ordinate system starting at 0 on the x- and y-axes and extending to 1.0 on the x-axis and 1.0 on the y-axis. You may now draw on this system. You can change the axis scaling using the “axis” command. But by default the scaling will increase to accommodate your objects.

6. Lines in Matlab Function line used to draw lines h = line(x, y) where x and y are x and y co-ordinates of the start and end of a line. H is a “handle” to the graphics object (used for setting properties. Example –x=[20 50] –y=[30 80] –hline = line(x,y) Draws a line from point x=20, y=30 to the point x=50, y=80

7. Rectangles in Matlab Function “rectangle” used to draw rectangles in Matlab. Often (in other packages and ‘C’ routines) rectangles are defined by 2 points only –Bottom left and top right. In Matlab –hrect = rectangle('Position', [100 100 50 25]) – (start position (x,y) then width and height.) Also used to draw ellipses and circles –hcirc = rectangle('Position', [100 100 50 25], 'Curvature', [1 1]). –A circle is an ellipse with the same (height as its width)

8. Polygons in Matlab h = patch(x, y, ‘c’) draws a polygon, the vertices of which are contained in x and y, and is filled by colour ‘c’. Example draw a red filled pentagon. –Need 5 points in x and y. –x=[50 25 100 175 150] –y=[50 100 150 100 50] –hpoly = patch(x, y, ‘r’)

9. Exercises Draw a line from point x=20, y=30 to the point x=50, y=60. Draw a rectangle with the bottom left hand corner at point x=30, y=50 and the top right hand corner at point x=70, y=80. Draw a circle of radius 3 and centred at point x=50, y=60. Draw a blue filled triangle.

10. Handles The main figure has a “handle” in the Matlab environment. Handles allow Matlab to keep track of figures and graphic objects. Within the main figure we have an axis object; this also also has a handle. It is a “child” object of the figure.

11. Handles We differentiate between or identify objects by their handles Sort of pointer. When we add drawing objects such as our lines, rectangles, patches, they become child objects of the axis object and are also identified by handles. So we now have an axes with four children. We can return there values using Hchild=get(haxes, ‘Children’)

12. Handles So our structure so far could be drawn as. Hfigure is the handle to the figure and has child object with handle haxes haxes has 4 children with handles hline, hrect, hcirc, and hpoly

13. Handles Or in hierarchical form Figure ‘children’ Axes linerectpatchrect Represents handles hfigure

14. Handles The handle of the figure is returned in variable “hfigure”. It gives us access to all the properties of the figure. get(h) returns a copy of the figure’s (object’s) properties including its children.

15. Handles, Try it Type “maindetails=get(hfigure)” maindetails lists all the properties in the figure. The structure includes handles to the child objects. We can use the handles to gain access to the child objects and alter their properties.

16. Properties, get() and set() We can retrieve a copy of the values associated with a graphic object through its handle by using S=get(hrect) The structure contains all the properties of the graphic object. However, since it is a copy we cannot change the actual information associated with the graphic object.

17. Properties, get() and set() So in true “OO’” style we must use an access method/function to adjust parameters.. Set(h,'PropertyName',Propert yValue) Get(h) or Get(h, ‘PropertyName’) returns the property. Note ‘quotes’

18. Order of objects As an drawing object is added to the axes object an entry (drawing object’s handle) is placed in the “children” array of the axis object. We can rearrange this array to change which object is on top. Again we are simply swapping handles We need to call “refresh” to see it.

19. Order of objects So we get the axes objects “children” array. hchild=get(haxes, ‘children’) Make a copy –htemp=hchild Rearrange the handles to the objects –htemp(4)=hchild(1) –htemp(1)=hchild(4) And set the axes “children: array to our new values set(haxes,’children’, htemp)

20. Deleting objects. We can delete an object using its handle. delete(hpoly) Better put it back! hpoly = patch(x, y, ‘r’)

21. Stroke, Fill and Colour All vector graphic shapes have stroke and fill “properties”. They affect how the graphic is drawn. Stroke is how lines (and outlines are drawn) fill is how shapes are filled in. One property is “colour” (‘color’ in Matlab. Some other properties for stroke are: –Width ‘LineWidth’ –style (dotted dashed etc.) ‘LineStyle’

22. Stroke, Fill and Colour To alter the fill and edge colour of a shape in Matlab: –set(hrect, 'FaceColor', [1 0 0]) for fill colour –set(hrect, 'EdgeColor', [1 0 0.5]) for stroke colour. –Where hr is a handle to the shape. Line width and style may also be applied to the shape’s outline.

23. Transparency and “alpha” channels. Another property of vector graphics is the ability to add transparency. Many packages allow the adjustment of transparency from 0% to 100%. An “alpha channel” (in addition to the colour channels id s provided with the objects for this purpose.

24. Transparency and “alpha” channels. We can access the alpha channel of our shapes by the alpha property of the “patch” drawing object in Matlab. Face and edge (fill and stroke) have separate alpha channels. It has values between 0 and 1. Defaults to 1 set(hpoly, ‘FaceAlpha’, 0.5)

25. Text Text may be added to vector graphics. htext=text(50,100, ‘Dogs and cats’) Properties “font” and “colour” (at least) may be changed.

26. Exercises Draw the rectangle as above but change the outline to red and fill the rectangle with cyan.

27. Curves Described mathematically. Polynomial equations. Degree of equations is the highest power of x. Linear y = ax + b –degree 1 Quadratic. y = ax 2 + bx + c. –degree 2 Cubic. y = ax 3 + bx 2 + cx + d –degree 3

28. Exercises Linear degree 1 e.g. y = 2x + 5. x = [-10 :10]; plot(x, 2*x + 5) Quadratic degree 2 e.g. y = x 2 + 0x + 5. x = [-10 : 10]; plot(x, (x.^2) + 5) Cubic degree 3 e.g. y = 2x 3 + 20x 2 + 3x +2. x = [-10 : 10]; plot(x, 2*(x.^3) + 20*(x.^2) + 3*x + 2)

29. Bezier curves P(t) = P 0 (1-t) 3 + P 1 3t(1-t) 2 + P 2 3t 2 (1-t) + P 3 t 3 As t takes values between 0 and 1. P’s are x, y points Really two equations. x(t) = x 0 (1-t) 3 + x 1 3t(1-t) 2 + x 2 3t 2 (1-t) + x 3 t 3. y(t) = y 0 (1-t) 3 + y 1 3t(1-t) 2 + y 2 3t 2 (1-t) + y 3 t 3

30. Bezier curves Consider t=0 then (1-t) = 1 –P(0) = (x 0, y 0 ) = P 0 so curve goes through first point. Consider t=1 then (1-t) = 0 –P(1) = (x 3, y 3 ) = P 3 so curve goes through last point. In between curve is pulled towards P 1 and P 2.,

31. Exercises Try Matlab function mybezier.m Examine the function. Set up four points P0, P1, P2, P3. –e.g. P0 = [2, 3] Run the function. Experiment with different points.

32. Other mathematical curves B-splines NURBS

33. Storage of Vector Graphics. Many formats –Autocad DXF. –Photoshop PSD. –Paintshop Pro PSP. We will consider PDF’s implementation because it may be viewed easily.

34. PDF format Portable Document Format Developed by Adobe. Developed from PostScript (and EPS (Encapsulated PostScript)). PostScript describes pages as vector graphics.

35. Graphics in PDF Similar operations to Matlab (and other). PDF allows many complex graphics operations. Lines Rectangles Curves No direct method for circles. We will look at a small selection to see that the PDF file can store vector graphics.

36. PDF graphic instructions. All PDF graphic instructions use postfix notation where parameters are followed by the graphics operation instruction. Fill and stroke parameters are set before the object is drawn. To see a created object we must “fill” and “stroke” it.

37. PDF graphic instructions. After adding to the PDF file we should really adjust the “Length” parameter of the stream and the “startxref” pointer at the end of the file. (see example file test.pdf) When counting additions a new line (in Windows/Dos) counts as 2 characters. However, Acrobat seems quite tolerant of errors in these fields. (Ghostscript is not)

38. Lines in PDF We must firstly move to the start of the path. (see test.pdf) –150 250 m Then we draw a line from this point to the end of line. –400 250 l To see the line we must draw the stroke –S Try your own using note pad saving the file by some different name.

39. Rectangles in PDF We set up the start point (x, y), then specify the width and height of the rectangle. Then use the re operator to create the rectangle. –200 300 50 75 re We must then draw the stroke and fill the rectangle. “B” does this in one operation. –B

40. Rectangles in PDF We set up the start point (x, y), then specify the width and height of the rectangle. Then use the re operator to create the rectangle. –200 300 50 75 re Creates a rectangle starting at point x=200 y=300 width=50, height = 75. We must then draw the stroke and fill the rectangle. “B” does this in one operation. –B

41. Bezier curve in PDF Set starting point (x, y) for curve using m operator –300 300 m Add the two “control points” (x, y pairs) and the end point and create the curve using the c operator: –300 400 400 400 400 300 c Draw and fill the curve. –The example also closes (joins the ends together) using the b operator (which also draws the stroke and fills the closed shape) –b

42. Stroke and fill in PDF Note that fill and stroke colours are set up before the affected object is drawn. In PDF terms we set the “graphics state”. Some “graphics state” commands: –Set line width Number w. –Set line style [ a b ] 0 d ( [ ] is solid line) –Set stroke colour R G B RG –Set fill colour R G B rg

43. Graphic manipulations Explained in detail in separate lecture. Translation, scaling, rotation. May be applied to vector or bitmap images. Less artefacts with vectors. This is set up as a “graphics state” in PDF using the cm operator.

44. Scaling Making objects larger or smaller In PDF we change the “current transformation matrix” of the “graphics state” using the cm operator. For scaling: – s x 0 0 s y 0 0 cm

45. Translation Moving objects left/right and up/down In PDF we change the “current transformation matrix” of the “graphics state” using the cm operator. For translation: – 1 0 0 1 t x t y cm

46. Rotation Making objects larger or smaller In PDF we change the “current transformation matrix” of the “graphics state” using the cm operator. For rotation: – cos  sin  -sin  cos  0 0 cm –Eg 0.707 0.707 –0.707 0.707 0 0 cm rotates the current co-ordinates by 45 o

47. Further information Hill, Computer Graphics. Prentice Hall. Adobe specifications available from www.adobe.com –Postscript –PDF