March 1, 2009 Dr. Muhammed Al-Mulhem1 ICS 415 Computer Graphics Rendering Geometric Primitives Dr. Muhammed Al-Mulhem March 1, 2009 Dr. Muhammed Al-Mulhem

March 1, 2009 Dr. Muhammed Al-Mulhem 1

ICS 415Computer Graphics

Rendering Geometric Primitives

Dr. Muhammed Al-MulhemDr. Muhammed Al-Mulhem

March 1, 2009March 1, 2009

Dr. Muhammed Al-MulhemDr. Muhammed Al-Mulhem

March 1, 2009March 1, 2009


Rendering geometric primitivesDescribe objects with points, lines, and surfaces Describe objects with points, lines, and surfaces

• Compact mathematical notationCompact mathematical notation

• Operators to apply to those representationsOperators to apply to those representations

Render the objectsRender the objects

• The rendering pipelineThe rendering pipeline

Reading:Reading:

• Appendix A1-A5Appendix A1-A5

Describe objects with points, lines, and surfaces Describe objects with points, lines, and surfaces

• Compact mathematical notationCompact mathematical notation

• Operators to apply to those representationsOperators to apply to those representations

Render the objectsRender the objects

• The rendering pipelineThe rendering pipeline

Reading:Reading:

• Appendix A1-A5Appendix A1-A5


Rendering

Generate an image from geometric primitivesGenerate an image from geometric primitivesGenerate an image from geometric primitivesGenerate an image from geometric primitives

Rendering

Geometric Primitives

Raster Image


3D Rendering Example

What issues must be addressed by a 3D rendering system?


Overview

3D scene representation3D scene representation

3D viewer representation3D viewer representation

Visible surface determinationVisible surface determination

Lighting simulationLighting simulation






Overview









How is the 3D scenedescribed in a computer?

How is the 3D scenedescribed in a computer?


3D Scene Representation

Scene is usually approximated by 3D primitivesScene is usually approximated by 3D primitives

• PointPoint

• Line segmentLine segment

• PolygonPolygon

• PolyhedronPolyhedron

• Curved surfaceCurved surface

• Solid object Solid object

• etc.etc.

Scene is usually approximated by 3D primitivesScene is usually approximated by 3D primitives

• PointPoint


• PolygonPolygon




• etc.etc.


3D Point

Specifies a locationSpecifies a locationSpecifies a locationSpecifies a location


3D Point

Specifies a locationSpecifies a location

• Represented by three coordinatesRepresented by three coordinates

Specifies a locationSpecifies a location

• Represented by three coordinatesRepresented by three coordinates

(x,y,z)


3D Vector

Specifies a direction and a magnitudeSpecifies a direction and a magnitudeSpecifies a direction and a magnitudeSpecifies a direction and a magnitude


3D Vector

Specifies a direction and a magnitudeSpecifies a direction and a magnitude

• Represented as the difference between two point positions Represented as the difference between two point positions by three coordinates (Vby three coordinates (Vxx, V, Vyy, V, Vzz).).

• Magnitude |V| = sqrt( VMagnitude |V| = sqrt( Vxx22 + V + Vyy

22 + V + Vzz22))

• Has no locationHas no location

Specifies a direction and a magnitudeSpecifies a direction and a magnitude

• Represented as the difference between two point positions Represented as the difference between two point positions by three coordinates (Vby three coordinates (Vxx, V, Vyy, V, Vzz).).

• Magnitude |V| = sqrt( VMagnitude |V| = sqrt( Vxx22 + V + Vyy

22 + V + Vzz22))

• Has no locationHas no location(Vx, Vy, Vz).(Vx, Vy, Vz).


Vector Addition/Subtraction

Operation u + v, with:Operation u + v, with:

• Identity Identity 00 : : vv + + 00 = = v v

• Inverse Inverse -- : : vv + (- + (-vv) = ) = 00

Addition uses the “parallelogram rule”:Addition uses the “parallelogram rule”:

Operation u + v, with:Operation u + v, with:

• Identity Identity 00 : : vv + + 00 = = v v

• Inverse Inverse -- : : vv + (- + (-vv) = ) = 00

Addition uses the “parallelogram rule”:Addition uses the “parallelogram rule”:

u+v

u

vu-v

uv

-v

-v


Vector Space

Vectors define a vector spaceVectors define a vector space

• They support vector additionThey support vector addition

– Commutative and associativeCommutative and associative

– Possess identity and inversePossess identity and inverse

• They support scalar multiplicationThey support scalar multiplication

– Associative, distributiveAssociative, distributive

– Possess identityPossess identity

Vectors define a vector spaceVectors define a vector space

• They support vector additionThey support vector addition

– Commutative and associativeCommutative and associative

– Possess identity and inversePossess identity and inverse

• They support scalar multiplicationThey support scalar multiplication

– Associative, distributiveAssociative, distributive

– Possess identityPossess identity


Affine Spaces

• Vector spaces lack position and distanceVector spaces lack position and distance

– They have magnitude and direction but no locationThey have magnitude and direction but no location

• Combine the point and vector primitivesCombine the point and vector primitives

– Permits describing vectors relative to a common locationPermits describing vectors relative to a common location

• A point and three vectors define a 3-D coordinate system A point and three vectors define a 3-D coordinate system

• Point-point subtraction yields a vectorPoint-point subtraction yields a vector

• Vector spaces lack position and distanceVector spaces lack position and distance

– They have magnitude and direction but no locationThey have magnitude and direction but no location

• Combine the point and vector primitivesCombine the point and vector primitives

– Permits describing vectors relative to a common locationPermits describing vectors relative to a common location

• A point and three vectors define a 3-D coordinate system A point and three vectors define a 3-D coordinate system

• Point-point subtraction yields a vectorPoint-point subtraction yields a vector


Coordinate Systems

Y

X

Z

Right-handedcoordinate

systemZ

X

Y

Left-handedcoordinate

system

Grasp z-axis with hand

Thumb points in direction of z-axis

Roll fingers from positive x-axis towards positive y-axis


Points + Vectors

Points support these operationsPoints support these operations

• Point-point subtraction: Point-point subtraction: QQ - - PP = = vv

– Result is a vector pointing fromResult is a vector pointing from PP toto QQ

• Vector-point addition: Vector-point addition: PP + + vv = = QQ

– Result is a new pointResult is a new point

• Note that the addition of two points is not definedNote that the addition of two points is not defined

Points support these operationsPoints support these operations

• Point-point subtraction: Point-point subtraction: QQ - - PP = = vv

– Result is a vector pointing fromResult is a vector pointing from PP toto QQ

• Vector-point addition: Vector-point addition: PP + + vv = = QQ

– Result is a new pointResult is a new point

• Note that the addition of two points is not definedNote that the addition of two points is not defined

P

Q

v


3D Line Segment

Linear path between two pointsLinear path between two pointsLinear path between two pointsLinear path between two points


3D Line Segment

Use a linear combination of two pointsUse a linear combination of two points

• Parametric representation:Parametric representation:

– P = PP = P11 + t (P + t (P22 - P - P11), (0 ), (0 t t 1) 1)

Use a linear combination of two pointsUse a linear combination of two points


– P = PP = P11 + t (P + t (P22 - P - P11), (0 ), (0 t t 1) 1)

P1

P2


3D Ray

Line segment with one endpoint at infinityLine segment with one endpoint at infinity

• Parametric representation: Parametric representation:

– P = PP = P11 + t V, (0 <= t < + t V, (0 <= t < ))

Line segment with one endpoint at infinityLine segment with one endpoint at infinity


– P = PP = P11 + t V, (0 <= t < + t V, (0 <= t < ))

P1

V


3D Line

Line segment with both endpoints at infinityLine segment with both endpoints at infinity


– P = PP = P11 + t V, (- + t V, (- < t < < t < ))

Line segment with both endpoints at infinityLine segment with both endpoints at infinity


– P = PP = P11 + t V, (- + t V, (- < t < < t < ))

P1

V


3D Line – Slope Intercept

Slope Slope =m =m

= rise / run= rise / run

Slope Slope = (y - y1) / (x - x1) = (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)= (y2 - y1) / (x2 - x1)

Solve for y:Solve for y:

y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1

or: y = mx + bor: y = mx + b

Slope Slope =m =m

= rise / run= rise / run

Slope Slope = (y - y1) / (x - x1) = (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1)= (y2 - y1) / (x2 - x1)

Solve for y:Solve for y:

y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1

or: y = mx + bor: y = mx + b

x

y

P2 = (x2, y2)

P1 = (x1, y1)

P = (x, y)


Euclidean Spaces

Q: What is the distance function between points and Q: What is the distance function between points and vectors in affine space?vectors in affine space?

A: Dot productA: Dot product

• Euclidean affine space = affine space plus dot productEuclidean affine space = affine space plus dot product

• Permits the computation of distance and anglesPermits the computation of distance and angles

Q: What is the distance function between points and Q: What is the distance function between points and vectors in affine space?vectors in affine space?

A: Dot productA: Dot product

• Euclidean affine space = affine space plus dot productEuclidean affine space = affine space plus dot product

• Permits the computation of distance and anglesPermits the computation of distance and angles


Dot Product

• TheThe dot productdot product or, more generallyor, more generally, , inner productinner product of two of two vectors is a scalar:vectors is a scalar:

vv11 • • vv22 = = xx11xx22 + + yy11yy22 + + zz11zz2 2 (in 3D)(in 3D)

• TheThe dot productdot product or, more generallyor, more generally, , inner productinner product of two of two vectors is a scalar:vectors is a scalar:

vv11 • • vv22 = = xx11xx22 + + yy11yy22 + + zz11zz2 2 (in 3D)(in 3D)

u θ

v


Dot Product

Useful for many purposesUseful for many purposes• Computing the length (Euclidean Norm) of a vector: Computing the length (Euclidean Norm) of a vector:

– lengthlength((vv) = ||) = ||vv|| = sqrt(|| = sqrt(v • v)v • v)

• NormalizingNormalizing a vector, making it unit-length: a vector, making it unit-length: v = v / ||v||v = v / ||v||

• Computing the angle between two vectors:Computing the angle between two vectors:

– u • v u • v = |= |uu| || |vv| cos(θ)| cos(θ)

• Checking two vectors for orthogonalityChecking two vectors for orthogonality

– u • v = 0.0u • v = 0.0

Useful for many purposesUseful for many purposes• Computing the length (Euclidean Norm) of a vector: Computing the length (Euclidean Norm) of a vector:

– lengthlength((vv) = ||) = ||vv|| = sqrt(|| = sqrt(v • v)v • v)

• NormalizingNormalizing a vector, making it unit-length: a vector, making it unit-length: v = v / ||v||v = v / ||v||

• Computing the angle between two vectors:Computing the angle between two vectors:

– u • v u • v = |= |uu| || |vv| cos(θ)| cos(θ)

• Checking two vectors for orthogonalityChecking two vectors for orthogonality

– u • v = 0.0u • v = 0.0 u θ

v


ProjectingProjecting one vector onto another one vector onto another

• If If vv is a unit vector and we have another vector, is a unit vector and we have another vector, ww

• We can project We can project ww perpendicularly onto perpendicularly onto vv

• And the result,And the result, u u, has length , has length w w • • vv

ProjectingProjecting one vector onto another one vector onto another

• If If vv is a unit vector and we have another vector, is a unit vector and we have another vector, ww

• We can project We can project ww perpendicularly onto perpendicularly onto vv

• And the result,And the result, u u, has length , has length w w • • vv

Dot Product

u

w

v

wv

wv

wvw

wu

)cos(


Dot Product

Is commutativeIs commutative

• u • v = v • uu • v = v • u

Is distributive with respect to additionIs distributive with respect to addition

• u • (v + w) = u • v + u • wu • (v + w) = u • v + u • w

Is commutativeIs commutative

• u • v = v • uu • v = v • u

Is distributive with respect to additionIs distributive with respect to addition

• u • (v + w) = u • v + u • wu • (v + w) = u • v + u • w


Cross Product

The The cross productcross product or or vector productvector product of two vectors is a of two vectors is a vector:vector:

The cross product of two vectors is orthogonal to bothThe cross product of two vectors is orthogonal to both

Right-hand rule dictates direction of cross productRight-hand rule dictates direction of cross product

The The cross productcross product or or vector productvector product of two vectors is a of two vectors is a vector:vector:

The cross product of two vectors is orthogonal to bothThe cross product of two vectors is orthogonal to both

Right-hand rule dictates direction of cross productRight-hand rule dictates direction of cross product

1221

1221

1221

tdeterminan222

11121 )(

y x y x

z x z x

z y z y

zyx

zyx

uuu zyx

vv


Cross Product Right Hand Rule• See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html

• Orient your right hand such that your palm is at the beginning of A and your fingers point in the direction of A

• Twist your hand about the A-axis such that B extends perpendicularly from your palm

• As you curl your fingers to make a fist, your thumb will point in the direction of the cross product

http://www.phy.syr.edu/courses/video/RightHandRule/index2.html


Cross Product Right Hand Rule

• See: http://www.phy.syr.edu/courses/video/RightHandRule/index2.html



























Other helpful formulas

Length = sqrt (x2 - x1)Length = sqrt (x2 - x1)22 + (y2 - y1) + (y2 - y1)22

Midpoint, p2, between p1 and p3Midpoint, p2, between p1 and p3

• p2 = ((x1 + x3) / 2, (y1 + y3) / 2))p2 = ((x1 + x3) / 2, (y1 + y3) / 2))

Two lines are perpendicular if:Two lines are perpendicular if:

• M1 = -1/M2M1 = -1/M2

• cosine of the angle between them is 0cosine of the angle between them is 0

• Dot product = 0Dot product = 0

Length = sqrt (x2 - x1)Length = sqrt (x2 - x1)22 + (y2 - y1) + (y2 - y1)22

Midpoint, p2, between p1 and p3Midpoint, p2, between p1 and p3

• p2 = ((x1 + x3) / 2, (y1 + y3) / 2))p2 = ((x1 + x3) / 2, (y1 + y3) / 2))

Two lines are perpendicular if:Two lines are perpendicular if:

• M1 = -1/M2M1 = -1/M2

• cosine of the angle between them is 0cosine of the angle between them is 0

• Dot product = 0Dot product = 0


3D Plane

A linear combination of three pointsA linear combination of three pointsA linear combination of three pointsA linear combination of three points

P1

P3P2


Origin

3D Plane

A linear combination of three pointsA linear combination of three points

• Implicit representation: Implicit representation:

– ax + by + cz + d = 0, or ax + by + cz + d = 0, or

– P·N + d = 0P·N + d = 0

• N is the plane “normal”N is the plane “normal”

– Unit-length vectorUnit-length vector

– Perpendicular to planePerpendicular to plane

A linear combination of three pointsA linear combination of three points

• Implicit representation: Implicit representation:

– ax + by + cz + d = 0, or ax + by + cz + d = 0, or

– P·N + d = 0P·N + d = 0

• N is the plane “normal”N is the plane “normal”

– Unit-length vectorUnit-length vector

– Perpendicular to planePerpendicular to plane

P1

N = (a,b,c)

d

P3P2


3D Sphere

All points at distance “r” from point “(cAll points at distance “r” from point “(cxx, c, cyy, c, czz)”)”

• Implicit representation:Implicit representation:

– (x - c(x - cxx))22 + (y - c + (y - cyy))22 + (z - c + (z - czz))22 = r = r 22


– x = r cos(x = r cos() cos() cos() + c) + cxx

– y = r cos(y = r cos() sin() sin() + c) + cyy

– z = r sin(z = r sin() + c) + czz

All points at distance “r” from point “(cAll points at distance “r” from point “(cxx, c, cyy, c, czz)”)”

• Implicit representation:Implicit representation:

– (x - c(x - cxx))22 + (y - c + (y - cyy))22 + (z - c + (z - czz))22 = r = r 22


– x = r cos(x = r cos() cos() cos() + c) + cxx

– y = r cos(y = r cos() sin() sin() + c) + cyy

– z = r sin(z = r sin() + c) + czz

r


3D Geometric Primitives

More detail on 3D modeling later in course More detail on 3D modeling later in course

• PointPoint


• PolygonPolygon




• etc.etc.

More detail on 3D modeling later in course More detail on 3D modeling later in course

• PointPoint


• PolygonPolygon




• etc.etc.

H&B Figure 10.46


Take a breath

We’re done with the primitivesWe’re done with the primitives

Now we’re moving on to study how graphics uses Now we’re moving on to study how graphics uses these primitives to make pretty picturesthese primitives to make pretty pictures

We’re done with the primitivesWe’re done with the primitives

Now we’re moving on to study how graphics uses Now we’re moving on to study how graphics uses these primitives to make pretty picturesthese primitives to make pretty pictures


Rendering 3D ScenesTransform

Illuminate

Transform

Clip

Project

Rasterize

Model & CameraModel & CameraParametersParameters Rendering PipelineRendering Pipeline FramebufferFramebuffer DisplayDisplay


Camera Models

The most common model The most common model is pin-hole camerais pin-hole camera • All captured light rays arrive along All captured light rays arrive along

paths toward focal point without paths toward focal point without lens distortion lens distortion (everything is in focus)(everything is in focus)

• Sensor response Sensor response proportional to radianceproportional to radiance

• Other models consider ...Depth of field, Motion blur, Lens distortion

The most common model The most common model is pin-hole camerais pin-hole camera • All captured light rays arrive along All captured light rays arrive along

paths toward focal point without paths toward focal point without lens distortion lens distortion (everything is in focus)(everything is in focus)

• Sensor response Sensor response proportional to radianceproportional to radiance

• Other models consider ...Depth of field, Motion blur, Lens distortion

View plane

Eye position(focal point)


Camera Parameters

What are the parameters of a camera?What are the parameters of a camera?What are the parameters of a camera?What are the parameters of a camera?


Camera ParametersPositionPosition

• Eye position (px, py, pz)Eye position (px, py, pz)

OrientationOrientation

• View direction (dx, dy, dz)View direction (dx, dy, dz)

• Up direction (ux, uy, uz)Up direction (ux, uy, uz)

ApertureAperture

• Field of view (xfov, yfov)Field of view (xfov, yfov)

Film planeFilm plane

• ““Look at” pointLook at” point

• View plane normalView plane normal

PositionPosition

• Eye position (px, py, pz)Eye position (px, py, pz)

OrientationOrientation

• View direction (dx, dy, dz)View direction (dx, dy, dz)

• Up direction (ux, uy, uz)Up direction (ux, uy, uz)

ApertureAperture

• Field of view (xfov, yfov)Field of view (xfov, yfov)

Film planeFilm plane

• ““Look at” pointLook at” point

• View plane normalView plane normal

right

back

Up direction

Eye Position

Viewdirection

ViewPlane

“Look at”Point


Moving the camera

View Frustum

Right

BackTowards

Up


The Rendering Pipeline

Transform

Illuminate

Transform

Clip

Project

Rasterize



Rendering: Transformations

We’ve learned about transformationsWe’ve learned about transformations

But they are used in three ways:But they are used in three ways:

• Modeling transformsModeling transforms

• Viewing transforms (Move the camera)Viewing transforms (Move the camera)

• Projection transforms (Change the type of camera)Projection transforms (Change the type of camera)

We’ve learned about transformationsWe’ve learned about transformations

But they are used in three ways:But they are used in three ways:

• Modeling transformsModeling transforms

• Viewing transforms (Move the camera)Viewing transforms (Move the camera)

• Projection transforms (Change the type of camera)Projection transforms (Change the type of camera)


The Rendering Pipeline: 3-D

Result:Result:

• All vertices of scene in shared 3-D “world” coordinate systemAll vertices of scene in shared 3-D “world” coordinate system

• Vertices shaded according to lighting modelVertices shaded according to lighting model

• Scene vertices in 3-D “view” or “camera” coordinate systemScene vertices in 3-D “view” or “camera” coordinate system

• Exactly those vertices & portions of polygons in view frustumExactly those vertices & portions of polygons in view frustum

• 2-D screen coordinates of clipped vertices2-D screen coordinates of clipped vertices

Scene graphObject geometry

LightingCalculations

Clipping

ModelingTransforms

ViewingTransform

ProjectionTransform



ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• All vertices of scene in shared 3-D “world” coordinate systemAll vertices of scene in shared 3-D “world” coordinate system



Modeling transformsModeling transforms

• Size, place, scale, and rotate objects and parts of the Size, place, scale, and rotate objects and parts of the model w.r.t. each othermodel w.r.t. each other

• Object coordinates -> world coordinatesObject coordinates -> world coordinates

Modeling transformsModeling transforms

• Size, place, scale, and rotate objects and parts of the Size, place, scale, and rotate objects and parts of the model w.r.t. each othermodel w.r.t. each other

• Object coordinates -> world coordinatesObject coordinates -> world coordinates

Z

X

Y

X

Z

Y



ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:

•All geometric primitives are illuminatedAll geometric primitives are illuminated


Lighting Simulation

Lighting parametersLighting parameters

• Light source emissionLight source emission

• Surface reflectanceSurface reflectance

• Atmospheric attenuationAtmospheric attenuation

• Camera responseCamera response

Lighting parametersLighting parameters

• Light source emissionLight source emission

• Surface reflectanceSurface reflectance

• Atmospheric attenuationAtmospheric attenuation

• Camera responseCamera responseN

N

Camera

Surface

LightSource


Lighting Simulation

Direct illuminationDirect illumination

• Ray castingRay casting

• Polygon shadingPolygon shading

Global illuminationGlobal illumination

• Ray tracingRay tracing

• Monte Carlo methodsMonte Carlo methods

• Radiosity methodsRadiosity methods

Direct illuminationDirect illumination

• Ray castingRay casting

• Polygon shadingPolygon shading

Global illuminationGlobal illumination

• Ray tracingRay tracing

• Monte Carlo methodsMonte Carlo methods

• Radiosity methodsRadiosity methods

NN

Camera

Surface

LightSource

N


ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• Scene vertices in 3-D “view” or “camera” coordinate systemScene vertices in 3-D “view” or “camera” coordinate system




Viewing transformViewing transform

• Rotate & translate the world to lie directly in front of Rotate & translate the world to lie directly in front of the camerathe camera

– Typically place camera at originTypically place camera at origin

– Typically looking down -Z axisTypically looking down -Z axis

• World coordinatesWorld coordinates View coordinatesView coordinates

Viewing transformViewing transform

• Rotate & translate the world to lie directly in front of Rotate & translate the world to lie directly in front of the camerathe camera

– Typically place camera at originTypically place camera at origin

– Typically looking down -Z axisTypically looking down -Z axis

• World coordinatesWorld coordinates View coordinatesView coordinates

Y

X

Z

Right-handedcoordinate

system


ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• Remove geometry that is out of viewRemove geometry that is out of view



ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• 2-D screen coordinates of clipped vertices2-D screen coordinates of clipped vertices




Projection transformProjection transform

• Apply perspective foreshorteningApply perspective foreshortening

– Distant = small: the Distant = small: the pinhole camerapinhole camera model model

• View coordinates View coordinates Screen coordinates Screen coordinates

Projection transformProjection transform

• Apply perspective foreshorteningApply perspective foreshortening

– Distant = small: the Distant = small: the pinhole camerapinhole camera model model

• View coordinates View coordinates Screen coordinates Screen coordinates


Perspective CameraPerspective Camera

Orthographic CameraOrthographic Camera

Perspective CameraPerspective Camera

Orthographic CameraOrthographic Camera



Rendering 3D Scenes

Transform

Illuminate

Transform

Clip

Project

Rasterize



Rasterize

Convert screen coordinates to pixel colorsConvert screen coordinates to pixel colorsConvert screen coordinates to pixel colorsConvert screen coordinates to pixel colors


Summary

Geometric primitivesGeometric primitives

• Points, vectorsPoints, vectors

Operators on these primitivesOperators on these primitives

• Dot product, cross product, normDot product, cross product, norm

The rendering pipelineThe rendering pipeline

• Move models, illuminate, move camera, clip, project to Move models, illuminate, move camera, clip, project to display, rasterizedisplay, rasterize

Geometric primitivesGeometric primitives

• Points, vectorsPoints, vectors

Operators on these primitivesOperators on these primitives

• Dot product, cross product, normDot product, cross product, norm

The rendering pipelineThe rendering pipeline

• Move models, illuminate, move camera, clip, project to Move models, illuminate, move camera, clip, project to display, rasterizedisplay, rasterize

Documents

March 1, 2009 Dr. Muhammed Al-Mulhem1 ICS 415 Computer Graphics Rendering Geometric Primitives Dr. Muhammed Al-Mulhem March 1, 2009 Dr. Muhammed Al-Mulhem