Fundamentals of Computer GraphicsPart 9

Discrete Techniques

prof.ing.Václav Skala, CSc.University of West Bohemia

Plzeň, Czech Republic

©2002Prepared with Angel,E.: Interactive Computer

Graphics – A Top Down Approach with OpenGL, Addison Wesley, 2001

Fundamentals of Computer Graphics 2

Discrete techniques

Polygons, lines and others primitives transformed, rasterized and displayed

API enables major mapping

• texture mapping uses pattern to be put on a surface of an object

• bump mapping – smooth surface is distorted to get variation of the surface

• environmental mapping (reflection maps) – enables ray-tracing like output

Fundamentals of Computer Graphics 3

Discrete techniquesAll three techniques rely on the map being stored as one-, two-

or three-dimensional digital image (!!Texture mapping on a graphics card is limited!!)

Causes ANTIALIASING errors

buffer – a memory block with spatial resolution n x m and with k bits

bitplane – single plane n x m with 1 bit only

pixel – picture element

Fundamentals of Computer Graphics 4

Texture MappingRegular patterns are mapped to an

object’s surface

Two dimensional texture T(s,t)s, t – texture coordinatesstored in texture memory as n x m array of texture elements - texels

Texture map associate a unique point of T with each point on a geometric object – mapped to screen coordinates

Fundamentals of Computer Graphics 5

Texture MappingDifficulties:

- mapping from texture to geometric coordinates

- 2D texture defined over a rectangular region in the texture space -> mapping to 3D region can be quite complex

- rendering based on pixel-to-pixel approach – the inverse mapping from screen coordinates to texture coordinates is needed

- because of shading – mapping areas-to-areas and not point-to-point is required -> antialising problems, moire patterns etc.

Fundamentals of Computer Graphics 6

Texture Mappingpixel (xs, ys) –

corresponds to(x, y, z) on “curved”object

problems with finding inverse mapping

difficulties with mapping

Fundamentals of Computer Graphics 7

Linear MappingMost curved surfaces represented parametrically as

p(u,v) = [x(u,v) , y(u,v) , z(u,v)]T

a point in the texture map T(s,t) is to be mapped to a point on the surface p(u,v) by a linear map

u = as + bt + c v = ds + et + f

(if ae bd mapping is invertible)

- mapping is easy to use

- it does not respectthe curvature of theobject

Fundamentals of Computer Graphics 8

Linear MappingLinear mapping isdefined as:

standard window-viewport approach.

)(

)(

minmaxminmax

minmin

minmaxminmax

minmin

vvtt

ttvv

uuss

ssuu

Fundamentals of Computer Graphics 9

Texture MappingTwo part mapping - steps

1. map the texture a simple 3Dintermediate surface – cylindercube, sphere etc.

2. the surface containing the mapped texture is mapped to the surface being rendered.

Can be applied in geometric or parametric coordinates

Fundamentals of Computer Graphics 10

Texture MappingSuppose a cylinder

x = r cos (2u)

y = r sin (2u)

z = v / h

and u,v (0,1) – then

s = u & t = v

we are able to map the texture WITHOUT distorting its shape

for a sphere the Mercator projection can be used

x = r cos (2u)

y = r sin (2u) cos (2v)

z = r sin (2u) sin (2v)

Fundamentals of Computer Graphics 11

Texture MappingThe second step is to map the texture values on the intermediate

object to the desired surface

Figure shows THREE possible strategies:

• texture value is projected on the surface in the normal direction

• inverse solutionfrom the surfacetexture elementis to be findaccording to thenormal

• if the center of the object is known – intersection with intermediate surface is computed and texture value assigned

Fundamentals of Computer Graphics 12

Texture Mapping in OpenGLOpenGL –

many mappingoptionsincluding 1D, 2D & 3D textures

GLubyte my_texels [512][512]; /* generated somehow */

glTexImage2D(GL_TEXTURE, 0, 3, 512, 512, 0,GL_RGB, GL_UNSIGNED_BYTE, my_texels);

/* 0 & 3 – level & components, 0 – border, format – see latter*/

glEnable(GL_TEXTURE_2D); /* enables texture mapping */

Fundamentals of Computer Graphics 13

Texture Mapping in OpenGL2D textures specification

glTexImage2D(GL_TEXTURE, level, components, width, height, border, format, type, array);

components (1 – 4 ) number of components RGBA to be affected with the map

level & border – parameters for fine control

Fundamentals of Computer Graphics 14

Texture Mapping in OpenGLglBegin(GL_QUAD);

glTexCoord2f(0.0, 0.0);

glVertex2f(x1, y1, z1);

glTexCoord2f(1.0, 0.0);

glVertex2f(x2, y2, z2);

glTexCoord2f(0.0, 1.0);

glVertex2f(x3, y3, z3);

glTexCoord2f(1.0, 1.0);

glVertex2f(x4, y4, z4);

glEnd ( );range of s, t changed to <0,0.5>

Fundamentals of Computer Graphics 15

Texture Mapping in OpenGLSTOP HERE!!!