Introduction to Computer Graphics CS 445 / 645 Lecture 6 Geometric primitives and the rendering pipepline Lecture 6 Geometric primitives and the rendering

Introduction to Computer GraphicsCS 445 / 645

Lecture 6Lecture 6

Geometric primitives andGeometric primitives andthe rendering pipeplinethe rendering pipepline

Lecture 6Lecture 6

Geometric primitives andGeometric primitives andthe rendering pipeplinethe rendering pipeplineM.C. Escher – Smaller and Smaller (1956)

Rendering geometric primitives

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

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

Appendix A1-A5Appendix A1-A5

H&B Figure 109

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

• Infinitely smallInfinitely small

Specifies a locationSpecifies a location


• Infinitely smallInfinitely small

(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


• Magnitude ||V|| = sqrt(dx dx + dy dy + dz dz)Magnitude ||V|| = sqrt(dx dx + dy dy + dz dz)

• Has no locationHas no location

Specifies a direction and a magnitudeSpecifies a direction and a magnitude


• Magnitude ||V|| = sqrt(dx dx + dy dy + dz dz)Magnitude ||V|| = sqrt(dx dx + dy dy + dz dz)

• Has no locationHas no location(dx,dy,dz)

Vector Addition/Subtraction

• operation operation u + vu + v, with:, with:

– Identity Identity 00 : : vv + + 00 = = v v

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

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

• operation operation u + vu + v, with:, 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

lGrasp z-axis with handlThumb points in direction of z-axis lRoll 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 Q: What is the distance function between points and vectors in affine space?and 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 Q: What is the distance function between points and vectors in affine space?and 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

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

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

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

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



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

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

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

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














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 Scenes

Transform

Illuminate

Transform

Clip

Project

Rasterize

Model & CameraModel & CameraParametersParameters Rendering PipelineRendering Pipeline FramebufferFramebuffer DisplayDisplay

Camera ModelsThe most common model is pin-hole camera The most common model is pin-hole camera

• All captured light rays arrive along paths toward focal point without lens distortion All captured light rays arrive along paths toward focal point without lens distortion (everything is in focus)(everything is in focus)

• Sensor response Sensor response proportional to radianceproportional to radiance

The most common model is pin-hole camera The most common model is pin-hole camera

• All captured light rays arrive along paths toward focal point without lens distortion All captured light rays arrive along paths toward focal point without lens distortion (everything is in focus)(everything is in focus)

• Sensor response Sensor response proportional to radianceproportional to radiance

Other models consider ...Depth of fieldMotion blurLens 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 plane Film 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 plane Film plane

• ““Look at” pointLook at” point

• View plane normalView plane normal

Eye Position

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

More on these methods later!More on these methods later!

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

ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:

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


Assignment 2

Due two and a half weeks from todayDue two and a half weeks from today

• Project description available onlineProject description available online

• We’ll discuss details in class on MondayWe’ll discuss details in class on Monday

Due two and a half weeks from todayDue two and a half weeks from today

• Project description available onlineProject description available online

• We’ll discuss details in class on MondayWe’ll discuss details in class on Monday

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

Introduction to Computer Graphics CS 445 / 645 Lecture 6 Geometric primitives and the rendering pipepline Lecture 6 Geometric primitives and the rendering