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## PowerPoint Slideshow about 'Object Modeling' - hermione-wright

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Presentation Transcript

Overview

- Introduction
- Object transformations
- Lines
- Planes
- Spheres
- Polygonal objects
- Hierarchical modeling
- OpenGL details

Introduction

- One step in the creation of a computer graphic image is the definition of the objects in the scene
- The shape of simple objects such as planes and spheres will be discussed in this chapter
- More complex curved surfaces will be discussed in the next chapter

Transformations

- Transformations are needed to:
- Position objects defined relative to the origin
- Build scenes based on hierarchies
- Project objects from three to two dimensions
- Transformations include:
- Translation
- Scaling
- Rotation
- Projections
- Transformations can be represented by matrices and matrix multiplication

Homogeneous Coordinates

- The three dimensional point (x, y, z) is represented by the homogeneous coordinate (x, y, z, 1)
- Some transformations will alter this fourth component so it is no longer 1
- In general, the homogeneous coordinate (x, y, z, w) represents the three dimensional point (x/w, y/w, z/w)

Translations

- The amount of the translation is added to or subtracted from the x, y, and z coordinates
- In general, this is done with the equations:

xn = x + tx

yn = y + ty

zn = z + tz

- This can also be done with the matrix multiplication:

Scaling

- Scaling increases or decreases the size of the object
- Scaling occurs with respect to the origin
- If the object is not centered at the origin, it will move in addition to changing size
- In general, this is done with the equations:

xn = sx * x

yn = sy * y

zn = sz * z

- This can also be done with the matrix multiplication:

Scaling

- Scaling can be done relative to the object center with a composite transformation
- Scaling an object centered at (cx, cy, cz) is done with the matrix multiplication:

Rotation

- Rotation can be done around any line or vector
- Rotations are commonly specified around the x, y, or z axis
- A positive angle of rotation results in a counterclockwise movement when looked at from the positive axis direction
- When rotating around an axis, the coordinate from that axis will not change

Rotation

- The matrix form for rotation around the x, y, and z axis are:
- x axis
- y axis
- z axis

Order of Transformations

- Matrix multiplication is not commutative so changing the order of transformation can change the result
- For example, changing the order of a translation and a rotation produces a different result:

Projection

- A projection transformation moves from three dimensions to two dimensions
- Projections occur based on the viewpoint and the viewing direction
- Projections move objects onto a projection plane
- Projections are classified based on the direction of projection, the projection plane normal, the view direction, and the viewpoint
- Two primary classifications are parallel and perspective

Parallel Projection

- All projectors run parallel and in the viewing direction
- Projecting onto the z = 0 plane along the z axis results in all z coordinates being set to zero

Parallel Projection

- Parallel projection can be done with the matrix multiplication:
- Parallel projections do not give visual clues as to distance (foreshortening)

Perspective Projection

- The viewpoint is the center of projection for a perspective projection
- With the viewpoint at the origin and the projection plane at z = d, a projector from (xO, yO, zO) intersects the plane d / zO along that line
- The intersection point is (xO * d / zO, yO * d / zO, zO * d / zO)

Perspective Projection

- In general, the intersection point is calculated by:

xn = x * d / z

yn = y * d / z

zn = z * d / z = d

- Homogeneous coordinates are needed to put this into matrix form:

Combining Transformations

- Matrices can be multiplied together to accomplish multiple transformations with one matrix
- A matrix is built with successive transformations occurring from right to left
- A combination matrix is typically built from the identity matrix with each new transformation added by multiplying it on the left of the current combination

Lines

- Not a significant component of scenes, but the basis for many parts of graphics
- The parametric equation for a line from P0 to P1 is:

L(t) = (1 – t) * P0 + t * P1

- As t takes on values from the range [0, 1], L(t) traces out the line between the two points
- An alternative form for this equation is:

L(t) = P0 + t * (P1 – P0) = P0 + t * v

- Where v is the vector P1 – P0

Planes

- A plane is defined by three points that are not co-linear
- Graphics is concerned with finite patches of these infinite planes
- Because the vertices of a triangular patch define the plane containing it, triangular patches are always planar
- The parametric form for a triangular patch is:

P(s, t) = (1 – s) * ((1 – t) * P0 + t * P1) + s * P2

Scan Converting a Triangular Patch

- The parametric equation for the sides can be used to identify where each row of pixels crosses the patch
- The orientation of the patch determines which edges get used for each part of the patch

Patch Rendering Problems

- Patches with more than three vertices may not be planar making interpolation inaccurate
- A better alternative is to break a larger patch into triangular pieces
- Interpolation of a non-triangular patch can also depend on orientation

Patch Rendering Problems

- A polygon is convex if a line connecting any two points inside the polygon is entirely in the polygon
- Scan conversion algorithms assume convex polygons
- Concave polygons can be broken down into convex pieces
- Triangular patches are always convex

Spheres

- Can be specified by giving the radius and the center location
- Locations on the surface of the sphere are given by the equation:

(x – xc)2 + (y – yc)2 + (z – zc)2 = r2

- Points on the silhouette of the sphere satisfy the equation:

(x – xc)2 + (y – yc)2 = r2

- Locations within this circle can be plugged back into the sphere equation to find the z coordinate for the location
- This process is very computationally expensive

Spheres

- One improvement on this process uses the eight-way symmetry of a circle
- Once one point on the circle has been found negating and swapping the values produces seven other values on the circle
- Though this is faster, the overall process is still slow because of squaring and squareroots

Spheres

- Bresenham produced an algorithm to determine locations on the perimeter of a circle with just increments and addition
- This algorithm can also be used to identify the locations on the front surface of a sphere

Polygonal Objects

- Objects with curved surfaces can be approximated with a collection of planar patches
- These patch collections can come from:
- A mathematical description of the surface
- A digitized version of the real object

Polygon Meshes

- Can be stored as an array of polygonal patches, but
- There will be a lot of duplicated data
- It is difficult to determine how patches are spatially related
- Changed object properties must be propagated to all of the patches

Triangular Meshes

- Data duplication can be reduced through the use of two lists, but patches still function independently
- A vertex list will store each vertex and its properties once
- A triangle list will give the index of the vertices for each patch

Winged-Edge Data Structure

- Data structure is built on the concept of an edge
- For each edge, the vertices, adjacent faces, and adjacent edges are stored

Level of Detail

- How detailed should an object model be?
- Level of Detail research uses human perception to answer this question
- Objects that will be small in the scene or may be moving fast in an animation may need less detail (i.e., fewer larger patches)

Hierarchical Modeling

- Objects are described in terms of their parts
- The parts may be further defined in terms of subparts
- Transformations can be applied to get subparts in the correct place and size within the part and the part in the correct place and size within the object

Hierarchical Modeling

- Low-level objects can be reused in the definition of a hierarchical model

Scene Graphs

- A scene graph is a type of hierarchical model
- Scene graphs are used in Java 3D

Hierarchical Modeling

- A stack of transformation matrices helps with hierarchical models
- As the program moves down the hierarchy, the current transformation is pushed before the new transformation is added
- As the program moves up the hierarchy, the top of the stack is used, which removes the transformations added lower in the hierarchy

OpenGL Coordinate Systems

- OpenGL uses a right-handed coordinate system
- The meaning of the coordinate values can vary by application
- The range of visible coordinates is determined by calls to glOrtho and glFrustum

The Grand, Fixed Coordinate System

- When transformations are given, they are applied to the local coordinate system of the object
- If a sequence of transformations are envisioned to get an object into the correct position within a grand and fixed coordinate system, those transformations are specified in the reverse order in OpenGL

Transformation Matrices

- OpenGL has two transformation matrices called the projection and model view matrices
- Only one of those is manipulated at a time
- glMatrixMode(GL_PROJECTION)
- glMatrixMode(GL_MODELVIEW)
- The glLoadIdentity() routine initializes the current matrix to the identity

Transformations

- The glRotatef and glRotated routines take four parameters:
- The angle
- The (x, y, z) values of the vector to rotate around
- The glTranslatef and glTranslated routines take three parameters representing the x, y, and z translation amount
- The glScalef and glScaled routines take three parameters representing the x, y, and z scale amount

Matrix Stacks

- OpenGL has a stack for the projection and model view matrices
- The projection matrix stack can hold at least two matrices
- The model view matrix stack can hold at least 32 matrices
- The glPushMatrix makes a copy of the the current matrix and pushed it onto the correct stack
- The glPopMatrix removes the top matrix from the stack and makes that the current matrix

Simple Objects

- Object vertices and their properties are specified between calls to glBegin and glEnd
- The parameter to glBegin indicates what is being specified
- GL_POINTS
- GL_LINES, GL_LINE_STRIP, GL_LINE_LOOP
- GL_POLYGON
- GL_TRIANGLE, GL_TRIANGLE_STRIP, GL_TRIANGLE_FAN
- GL_QUAD, GL_QUAD_STRIP

Points

- The vertices specified between the begin and end are drawn as individual points

Lines

- Pairs of vertices are used to draw separate lines for GL_LINES
- Lines are drawn between every pair of vertices for GL_LINE_STRIP
- A line is also drawn between the first and last vertex for a GL_LINE_LOOP
- Any extra vertices are ignored

Polygons

- The area surrounded by the lines defined by the vertices is drawn filled in based on other OpenGL parameters

Triangles

- Each set of three vertices is drawn as a triangular polygon for GL_TRIANGLES
- Each extra vertex adds a new triangular patch using the previous two vertices for GL_TRIANGLE_STRIP
- The first vertex is used with each other pair of vertices for GL_TRIANGLE_FAN
- Any extra vertices are ignored

Quads

- Each set of four vertices is drawn as a four-sided patch for GL_QUADS
- Each extra pair of vertices adds a new four-sided patch using the previous two vertices for GL_QUAD_STRIP
- Any extra vertices are ignored

GLUT Objects

- The GLUT provides routines to draw a set of objects
- glutWireSphere and glutSolidSphere
- Radius, slices, stacks
- glutWireCube and glutSolidCube
- Size
- glutWireTorus and glutSolidTorus
- Inner radius, outer radius, sides, rings
- glutWireCone and glutSolidCone
- Base radius, height, slices, stacks

GLUT Objects

- glutWireIcosahedron and glutSolidIcosahedron
- glutWireDodecahedron and glutSolidDodecahedron
- glutWireOctahedron and glutSolidOctahedron
- glutWireTetrahedron and glutSolidTetrahedron
- glutWireTeapot and glutSolidTeapot
- Size

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