Graphic Designing

Polygon Meshes- Important Steps

2011-09-22T02:30:00.001-07:00

Polygon Meshes- Important Steps

When working with polygon meshes, it is important to ensure that the mesh representation is consistent.

– All polygons are closed.

– Each edge is used at least once and not more than twice.

– Each vertex is referenced by at least two edges

Polygon Meshes - Pointers to an edge list

2011-09-22T02:28:00.001-07:00

Polygon Meshes - Pointers to a vertex list

2011-09-22T02:26:00.001-07:00

Polygon Meshes

2011-09-16T23:10:00.001-07:00

A polygon mesh is a set of bounded polygons (vertices, edges, and faces) such that each edge is shared by at most 2 polygons

Polygon meshes can be represented in several ways

– Explicit representation

P={(x1,y1,z1),........,(xn,yn,zn)}

Surface Modeling | Graphics Designing

2011-09-16T23:06:00.002-07:00

Surface Modeling

Several methods are available to model surfaces

– Polygon meshes
• Surface is represented as a set of bounded planar surfaces
– Parametric polynomial curves
• Surface is defined by 3 equation in a parameter, t, one for each principal axis (x, y, z)

– Parametric bivariate polynomial surface patches
• Similar to parametric polynomial curves but this time each equation has two variables
– Quadratic surfaces
• Surface is implicitly defined by an equation f(x, y, z) = 0

Yaw, Pitch and Roll

2011-08-02T22:01:00.002-07:00

In flight simulators, the angles of rotation are about the center of mass of the airplane. The camera is in the nose of the plane.

Pitch :

Roll :

Yaw :

The U-V-N System

2011-08-02T21:53:00.002-07:00

The camera plane is defined by 3 parameters:

View Reference Point (VRP): A point (in the world frame) indicating the center of the camera's image plane. This specifies the position of the camera in space.
View Plane Normal (VPN): A unit vector perpendicular to the camera's image plane. This specifies the orientation of the camera.
View Up (VUP): A vector indicating the up direction (in world coordinates) for the camera.

3d Viewing in Computer Graphics

2011-08-02T21:46:00.003-07:00

3d Viewing in Computer Graphics

Two stages to the 3D to 2D transformation

First a clipping,

Then a projection.

The view volume of a perspective projection is the infinite volume defined by the combination of the PRP and the view window, which defines a pyramid. This can be contrasted to the view volume for a parallel projection which gives an infinite parallelepiped.

Projection Matrices

2011-08-02T21:38:00.002-07:00

Projection Matrices

Hierarchy of Projections

2011-07-30T21:51:00.000-07:00

Perspective Projections | OpenGL

2011-07-28T01:14:00.002-07:00

Perspective Projections

1. The visual effect of the perspective projection is similar to the human visual system known as perspective foreshortening i.e. the size of perspective projection object varies inversely with the distance of the object from COP.

2. Perspective projection does not preserve parallel lines.

3. Vanishing Point
Any set of parallel lines that are not parallel to projection plan appear to converge to a point called vanishing point.

* In this case lines parallel to x and y do not converge, only lines parallel to z do so

* One point perspective projection of a cube

* A two point perspective projection of a cube

Parallel Projections | OpenGL

2011-07-26T22:14:00.002-07:00

Parallel Projections

In case of parallel projection instead of having COP we have direction of projection (DOP), so the perspective projection whose COP is at infinity becomes parallel projection.
In parallel projections, the lines projecting from an object onto the image plane are parallel.
Two parallel projection systems:
1. Orthographic projection

2. Oblique projection

In these the projection plane is normal and direction of projection differ.

Its major types are:
– Cabinet
– Cavalier

Projections

2011-07-18T03:06:00.002-07:00

Projection
Projection is the change in coordinate system and that is usually from 3D to 2D.

Projectors
The projection of a 3D object is described by straight `projection rays' (called projectors).
Types Of Projection

Projections can be divided into two basic
types
1. Perspective
2. Parallel
• The distinction between two is on the relation of COP and projection plane.
• If distance of one projector to another is finite then it is perspective projection otherwise it is parallel.

Perspective Projection

Parallel Projection

Viewing in Computer Graphics

2011-07-18T02:56:00.002-07:00

Viewing in Computer Graphics

Viewing refers to positioning the camera in a coordinate frame and projection of the 3D object onto the 2D image plane of the camera.

Transformations Change Coordinate Systems

2011-07-18T02:53:00.000-07:00

Transformations Change Coordinate Systems

One useful coordinate system transformation is given by:

Which transforms from a left handed to a right handed coordinate systems (and is its own inverse)

3D Homogenous Transformations

2011-07-18T02:45:00.001-07:00

3D Homogenous Transformations
1. Rotation about x-axis

2. Rotation about y-axis

3. General form of a composition of transformations

3D Homogenous Transformations

2011-07-18T02:13:00.001-07:00

3D Homogenous Transformations

1. Translation

2. Scaling

3. Rotation

Rotation Direction

2011-07-18T02:01:00.001-07:00

Rotation Direction

Which way is +ive rotation

1. Look in –ve direction (into +ve arrow)

2. Counter Clock Wise is +ve rotation

3D Coordinate Systems | OpenGL

2011-07-18T01:56:00.001-07:00

1. Right hand coordinate system

2. Left hand coordinate system(

Not used in OpenGL)

Composing Transformation

2011-07-16T12:04:00.001-07:00

Applying several transforms in succession to form one overall transformation is known as Composing Transformation.

(M3 x (M2 x (M1 x P ))) = M3 x M2 x M1 x P

1. Transformation products may not be commutative.

A x B != B x A

2. Some cases where A x B = B x A

A B

translation translation

scaling scaling

rotation rotation

uniform scaling rotation

(sx = sy)

3. Rotation and translation are not commutative.

Example :

Suppose we wish to reduce the square in the following image to half its size and rotate it by 45° about point P.

We need to remember that that scaling and rotation takes place with respect to the origin.

1– Translate square so that the point around which the rotation is to occur is at the origin

2– Scale

3– Rotate

4– Translate origin back to position P

Window to Viewport Transformation

2011-07-10T05:51:00.001-07:00

To display the appropriate images on the screen (or other device) it is necessary to map from world coordinates to screen or device coordinates.

This transformation is known as the window to viewport transformation, the mapping from the world coordinate window to the viewport (which is given in screen coordinates)

In general the window to viewport transformation will involve a scaling and translation as in figure below here the scaling in non-uniform (the vertical axis has been stretched).

Computer Graphics and Geometric Transformations

2011-04-29T00:07:00.001-07:00

2D Transformations
2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models.There are three types of transformations that are essential in computer graphics

1-Translation
Moving an object to a new location by adding to objects x and y co-ordinates.
2-Scaling
Changing the size of the object.This can be uniform where both dimensions are re-sized by same factor or non-uniform.
3-Rotation
Object is rotated around the origin by specified angle.

Homogeneous Transformations
In order to represent a translation as a matrix operation we use 3 x 3 matrices and pad our points to become 1 x 3 matrices.

A program (Objects, Lights, Camera)

2011-04-27T23:57:00.002-07:00

Lighting

2011-04-27T23:50:00.001-07:00

Direction light source
Position light source
glLightfv( Light#, Attribute, …)
GLfloat position[] = {10, 10, 10, W} glLightfv(GL_LIGHT0, GL_POSITION, position) If (W) is zero the position is treated as a direction (a 1x3 vector); otherwise, it is treated as a position (a 1x4 vector)
glEnable(GL_LIGTHING)
glEnable(GL_LIGHT0)

Drawing in 3D

2011-04-27T23:48:00.001-07:00

Depth buffer
   Allows scene to remove hidden surfaces. Use glEnable(GL_DEPTH_TEST) to enable it.
   1-glPolygonMode( Face, Mode )
       Face: GL_FRONT, GL_BACK, GL_FRONT_AND_BACK
      Mode: GL_LINE, GL_POINT, GL_FILL
2-glCullFace( Mode )
      Mode: GL_FRONT, GL_BACK, GL_FRONT_AND_BACK
3-glFrontFace( Vertex_Ordering )
      Vertex Ordering: GL_CW or GL_CCW