compound transformations, geometry, lighting and shadinglarsvmag/iti43309_13/lecture03.pdf ·...

Compound Transformations, Geometry, Lightingand Shading

Lars Vidar Magnusson

September 2, 2013

http://www.it.hiof.no/~larsvmag/iti43309_13/index.html

Lars Vidar Magnusson Computer Graphics 2013

http://www.it.hiof.no/~larsvmag/iti43309_13/index.html

Rows and Columns

Matrices can be represented in two ways, either as column-majoror as row-major. This is purely a notational difference, in that thesame result is acheived by changing either the multiplication orderor the transposition.

v ′ = Mv

v ′T = vTMT

The above order is considered post-multiplication order, which isthe standard in OpenGL. Direct3D uses pre-multiplication order.

v ′ = vM


Compound Transformations

Two transformation matrices can be combined into one bymultiplying them together. In OpenGL matrix multiplication, whichinclude any of the transformation functions, is done in post-order.

M′ = MM1

A vertex is transformed by compound transformation in thefollowing way.

v ′ = M1M2M3M4v

Note then that the matrices can be perceived as being applied inthe reverse order.


Storing and Restoring Matrices in OpenGL

OpenGL uses internal matrices to represent the current modelview,projection and texture matrices. These matrices can be stored on astack using the functions glPushMatrix and glPopMatrix, whichstores and retrieves the current matrix respectively.

Figure 1: Pushing and popping the modelview matrix can helphierarchical rendering


Geometry

Complex models are typically represented by a meshes/polyhedrain todays three-dimensional APIs and hardware. Each mesh iscomposed of the following atomic elements:

Vertices

Edges


Standard Primitives

There are a number of standard primitives which build on top ofthe vertices and edges in todays three-dimensional APIs.

Points (just the vertices)

Lines

Triangles

Quads

Polygons

The hardware available have been optimized for triangle rendering,so quads and polygons are typically converted before they aredrawn.


Primitive Rendering In OpenGL

The mentioned primitives can all be rendered in OpenGL using thefollowing set of functions:

g l B e g i n ( mode ) ;g l V e r t e x f ( x , y , z ) ;g lEnd ( ) ;

The parameter mode specify what kind of primitive to draw.

GL POINTS

GL LINES

GL TRIANGLES

GL QUADS

GL POLYGON

...


Primitive Modes in OpenGL

The different primitive drawing modes in OpenGL specify how thepoints, if at all, the vertices are connected.

Figure 2: GL POINTS turns eachvertex into a point

Figure 3: GL POLYGON turns allthe vertices into a face


Primitive Modes in OpenGL Continued

Figure 4: GL QUADS turns 4vertices into a face

Figure 5: GL TRIANGLES turns3 vertices into a face

Figure 6: GL QUAD STRIP thevertices into strip of quads

Figure 7: GL TRIANGLE STRIPthe vertices into strip of triangles


General Data Structures

The primitive modes offer a method for drawing simple primitives,but it is also possible to use a more general approach. This is doneusing an array approach.

Vertex attributes

PositionNormalTextures coordinate...

Indices

There are at least two ways of using general arrays in OpenGL, seevertex arrays and vertex buffer objects.


GLU Quadrics

The GLU library offers a simple way of drawing simples shapes,such as cylinders and spheres.

Figure 8: A cylinder drawn withgluCylinder(q, br, tr, h, sl, st)

Figure 9: A sphere drawn withgluSphere(q, r, sl, st)


Lighting and Materials

OpenGL (as well as Direct3D) uses a gouraud shading model alongwith a Blinn-Phong reflection model. The lighting model involvefour different kinds of light: ambient, diffuse, specular and emitted.


Emitted Lighting

Emitted lighting is light emitted from a surface in the scene, andas such it depends on nothing else than the material.

Ier = Mer

Ieg = Meg

Ieb = Meb

Like ambient lighting, the emitted light is a ”flat” light in that theshape of the model is neglected.


Ambient Lighting

Ambient light is cast uniformly on all objects in the scene, due tonot having a direction. The resulting intensity Ia of each of thecolor components are calculated by combining the light La with thematerial Ma:

Iar = LarMar

Iag = LagMag

Iab = LabMab

The ambient light component can both be set globally and perlight source.


Diffuse Lighting

Diffuse lighting, unlike ambient, has direction, and it can eitherhave a specified source or be cast from a source infinitely far away.The direction makes it possible to alter the outcome according tohow the light hits a surface.The intensity Id are calculated by incorporating the angle αbetween the surface normal N and the diffuse light direction L.

Idr = LdrMdr cosα = LdrMdr (N · L)

Idg = LdgMdg cosα = LdgMdg (N · L)

Idb = LdbMdb cosα = LdbMdb(N · L)

Note that only angles between 0 and 90 are of interest, and thatthe vectors L and N has to be of unit length.


Specular Lighting

Specular lighting aims to emulate the shininess of surfaces. Thismakes it dependent not only on the light direction L, but also onthe position of the eye.

Figure 10: A diagram of the elements of specular lighting

Isr = LsrMsr cossh β = LsrMsr (R · V )sh

Isg = LsgMsg cossh β = LsgMsg (R · V )sh

Isb = LsbMsb cossh β = LsbMsb(R · V )sh


Weakening of Light

The lighting model allows for weaking of light specified by anattenuation factor fatt . This factor can be calculated using thedistance to the light source d and a set of constants.

fatt =1

c1 + c2d + c3d2

c1 specify constant attenuation

c2 specify linear attenuation

c3 specify quadric attenuation


Lighting in OpenGL

We have covered the general math behind OpenGL’s lightingmodel, but now it is time to look at how to do lighting using theOpenGL API.

g l E n a b l e ( GL LIGHTING ) ;g l E n a b l e ( GL LIGHTn ) ;g l D i s a b l e ( GL LIGHTn ) ;g l D i s a b l e ( GL LIGHTING ) ;

These simple statements enable and disable lighting and a specificlight; n, which could range from 0 to 7.


Lighting in OpenGL Continued

Ambient lighting is controlled using:

g l L i g h t ( lno , GL AMBIENT , Lar , Lag , Lab , a l p h a )g l M a t e r i a l ( s i d e , GL AMBIENT , Mar , Mag , Mab , a l p h a )

Diffuse lighting is controlled using:

g l L i g h t ( lno , GL DIFFUSE , Ldr , Ldg , Ldb , a l p h a )g l M a t e r i a l ( s i d e , GL DIFFUSE , Mdr , Mdg , Mdb, a l p h a )

Specular lighting is controlled using:

g l L i g h t ( lno , GL SPECULAR , Lsr , Lsg , Lsb , a l p h a )g l M a t e r i a l ( s i d e , GL SPECULAR , Msr , Msg , Msb , a l p h a )g l M a t e r i a l f ( s i d e , GL SHININESS , sh ) ;

Emitted lighting is controlled using:

g l M a t e r i a l ( s i d e , GL EMISSION , Mer , Meg , Meb , a l p h a )


Light Attributes in OpenGL

We can specify the position of a light source using:

g l L i g h t ( lno , GL POSITION , x , y , z , w)

The three attenuation constants can be modified using:

g l L i g h t f v ( lno , GL CONSTANT ATTENUATION, c1 )g l L i g h t f v ( lno , GL LINEAR ATTENUATION , c2 )g l L i g h t f v ( lno , GL QUADRATIC ATTENUATION , c3 )

The spot cutoff and exponent can be set using:

g l L i g h t ( lno , GL SPOT CUTOFF , v )g l L i g h t ( lno , GL SPOT EXPONENT , e )


Shading and Smoothing

Shading represent how light affect th surfaces in a scene. OpenGLuse a technique called gouraud shading which is a simplification ofphong shading. Gouraud shading works by calculating theintensities at each vertex, and then do a interpolation to determinethe intensities on the remaining surface.


Gouraud Shading

Gouraud shading works by doing a linear interpolation between theintensities of the vertices.

Figure 11: A diagram illustrating gouraud shading


Gouraud Shading Continued

The diagram on the previous slide declares I1, I2 and I3 as theintensities of the vertices of triangle. The intensity Ia, Ib andultimately I can then be calculated as follows:

y1 − yay1 − y2

=I1 − IaI1 − I2

Ia =ya − y2y1 − y2

I1 +y1 − yay1 − y2

I2

y3 − yby3 − y2

=I3 − IbI3 − I2

Ib =yb − y2y3 − y2

I3 +y3 − yby3 − y2

I2

I =xb − x

xb − xaIa +

x − xaxb − xa

Ib


Gouraud Shading in OpenGL

Gouraud shading can be turned on or off as follows:

glShadeModel (GL SMOOTH ) ;g lShadeModel ( GL FLAT ) ;

Figure 12: The difference between GL FLAT and GL SMOOTH


Limitations of Gouraud Shading

There are some limitations in the gouraud shading model that oneshould be aware of.

If a light source hits a surface without touching a singlevertex, the effect will not be seen.

Rough models will get a jittered effect due to the limitationsof linear interpolation of the intensities.

Figure 13: The limitations of gouraud shading


compound transformations, geometry, lighting and shadinglarsvmag/iti43309_13/lecture03.pdf ·...

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