The secret math behind

Transcription

1 he secret math behind modern computer graphics b Magnus Ranlöf and Johan Winell ma 99 Preface D - graphics Computer graphic ma seem to become more and more complicated, but the fact is, it is controlled b rather simple geometr and linear algebra and has alwas been. he major reason wh computer graphics improve all the time is because of the higher performances of computers. B reading the following ou will hopefull increase our understanding of computer graphics and the secrets behind it. his article will take ou on a journe from the simplest concepts of D-graphics, through D-graphic representation, to the dealing with shading and perspective problems. No pre-knowledge in computer graphics is needed, but some knowledge in fundamental mathematics and geometr ma be useful. D object representation D objects are represented b an amount of points in a two dimensional co-ordinate sstem. he points in an object are declared in a matri. For D objects the matri has the N dimension. N If lines are drawn between the points a figure will appear. Contents: D graphics - Object representation - ransformation - Scaling - ranslation - Rotation Homogenous co-ordinates D graphics - Object representation - ransformation of a D - object - Depth problems - Back face removal - Lightning and shading - Perspective uthors words D - transformation Fortsättning på sidan he following transformation can be done to a D object. Scaling, to make the object smaller or bigger. ranslation, to move the object in the coordinate sstem. Rotation, to rotate the object in the coordinate sstem.

2 hese transformations are applied to the object b the use of matri operations. he object matri is multiplied or added to a transformation matri. Scaling o obtain a scaling of the object ou must multipl to a scaling matri, S. he matri obtained after the multiplication is a scaled object,. and. he result will be the matri object moved units in the direction, and unit in the direction. S S S he value for S and S will double the size of the object. he value wouldn t change the object at all. With the scaling matri, one can easil scale just the ais, or the -ais independent of each other. If the -object were to be plotted it would be twice the size of the object. Rotation o rotate a D-object ou must multipl a rotation matri, R, to our original matri,. he rotation matri is defined as follows: R is the rotation angle, defined counter clockwise. o rotate the object, perform the following operation. R If the rotation angle is 9 degrees, the result will be as follows: ranslation ranslation of an object is obtained b adding a translation matri,, to the original matri,. he resulting matri will be translated object called. Note that the object is rotated around the origin (,). he following -matri has the values

3 Homogenous coordinates o be able to do these transformations in a sequence onl involving multiplication, homogenous co-ordinates is the easiest wa to accomplish it. In the two-dimensional case, an additional row filled with a is introduced to the object description. o simplif it let us consider a gle point. (See eample below) It is now possible to perform translation with a matri multiplication, it earlier involved addition to each point. o be able to perform the multiplication the transformation matri has to be modified. (See eample below) translation can now be done like this: rotation for eample will look like this: With the use of homogenous co-ordinates, it is now possible to perform composite transformations b ug onl one epression. (E g) a translation followed b a rotation: he epression is eecuted backwards, e g the rotation matri is performed first and the translation matri is performed after. o rotate an object around an arbitrar point, first translate the point to the centre of rotation (the origin), rotate the object and translate back the point. he composite epression, C, is defined as C back R,where is the translation, R the rotation and back the translation back. D-graphics D-object representation ll objects including D- and D- objects can be described b man polgon surfaces. he triangle is a ver useful shape b which boes and spheres easil can be described. (Imagine a mirror ball, which contains man thousands of quadratic mirrors.) When dealing with D-objects a third co-ordinate, Z, must be considered. he D-co-ordinate sstem will look as follows:

4 ransformation of a D-object ransformations of D-objects are ver similar to transformations of D-objects, the onl thing that distinguishes is that a third dimension is added. he following matries describes a, ) scaling, B) translation, and C) rotation. S ) S S z B) ) C z When describing a triangle area in a D-object, three points must be specified, each with three co-ordinates, an -, a - and a Z-co-ordinate. he object description for one triangle can be described like this ug homogenous coordinates. Each of them is described with three corners (vertees). (,,7)(,,) (,,9)or as a matri : 7 9 ) he matri represents a Z-ais rotation. Note that the Z-co-ordinates are left unchanged. Depth problems When looking at a wireframe it is impossible to decide which side is at the front and which is at the back. pramid can be described b four triangles. Since the triangles in a pramid have connecting edges, some of the corners will be the same. herefore it is efficient to put all the corners in a gle matri: Specifing three specific corners of the amount of corners can now make the definition of a triangle. o draw a wireframe of an object, its different triangles must be drawn. o make this it is just to specif the three corners of the triangle, and draw lines between them. One wa to create an impression of depth in the wireframe, without the use of perspective, is ug a method called depth cueing. he lines in the front are drawn with higher intensit than the lines in the back. he use of this method makes the rear lines look shaded, compared to the front lines. When drawing solid objects a similar problem occurs: Some surfaces of the object are not to be shown to the viewer, because of their placement in the back of the object. o present the object correctl ou must either draw the surfaces in a certain order, or make sure that the surfaces in the rear never are drawn. he first option can be ver difficult to handle, because the drawing order is described in the object representation.

5 Back face removal he eas wa to get around this problem is to set a visibilit flag for each surface. he surfaces in the front are given a flag show and the surfaces in the back are given a do not show flag. When the solid is to be drawn, onl the surfaces with the show flag are drawn. he actual result is that surfaces, which represent the rear, are not shown for the viewer. he flags are set b calculating the scalar product between each triangles normal vector and a vector which represents the direction of the viewer. If the result is negative the normal points awa from the viewer, the do not show flag is set, and the surface is not shown. Which side is the front? Lightning and shading o create a realistic picture, it is important to use shades, and other lightning effects in a proper wa. First, set the co-ordinates of the light source, and construct a vector between the light source and the mid point of the object. hen calculate the normal of the surface. he scalar product between the light source vector and the normal of the illuminated surface is the light intensit in that particular surface. represents the highest intensit (he normal and the light source vector has the same direction. represents the lowest intensit (he normal has the opposite direction). he intensit value can be higher than one and less than zero. Values eceeding one, is set to one. negative value is taken care of b the back face removal procedure. then we were dealing with depth cueing. he other wa to create a sense of depth is b the use of perspective. When drawing a perspective object, the object is scaled depending on its distance to the viewer. One wa to implement the scaling is to use the last co-ordinate in an object, described in homogenous co-ordinates, and let that coordinate represent the scaling. ll other coordinates are divided b the scaling value. uthors words: his wasn t so hard after all, was it? If ou had an problems with the linear algebra, Krpa gå, b the famous Swedish mathematician Peter Hackman, will give ou all the knowledge ou need. Hopefull this brief tet increased our appetite for more knowledge in computer graphics. We strongl recommend ou to read another pages about it in Computer graphics second edition b Donald Hearn and M.Pauline Baker. It is a little bit tougher to read, but it will widen our knowledge in the topic. Perspective Depth in a picture has been discussed before, but