CRTs – A Review CRT technology hasn’t changed much in 50 yearsCRT technology hasn’t changed much in 50 years Early television technologyEarly television

CRTs – A Review

• CRT technology hasn’t changed much in 50 yearsCRT technology hasn’t changed much in 50 years

• Early television technologyEarly television technology

– high resolution high resolution

– requires synchronization between video signal and requires synchronization between video signal and electron beam vertical sync pulseelectron beam vertical sync pulse

• Early computer displaysEarly computer displays

– avoided synchronization using ‘vector’ algorithmavoided synchronization using ‘vector’ algorithm

– flicker and refresh were problematicflicker and refresh were problematic

• CRT technology hasn’t changed much in 50 yearsCRT technology hasn’t changed much in 50 years

• Early television technologyEarly television technology

– high resolution high resolution

– requires synchronization between video signal and requires synchronization between video signal and electron beam vertical sync pulseelectron beam vertical sync pulse

• Early computer displaysEarly computer displays

– avoided synchronization using ‘vector’ algorithmavoided synchronization using ‘vector’ algorithm

– flicker and refresh were problematicflicker and refresh were problematic

CRTs – A Review• Raster Displays (early 70s)Raster Displays (early 70s)

– like television, scan all pixels in regular patternlike television, scan all pixels in regular pattern

– use frame buffer (video RAM) to eliminate sync problemsuse frame buffer (video RAM) to eliminate sync problems

• RAMRAM

– ¼ MB (256 KB) cost $2 million in 1971¼ MB (256 KB) cost $2 million in 1971

– Do some math…Do some math…

- 1280 x 1024 screen resolution = 1,310,720 pixels1280 x 1024 screen resolution = 1,310,720 pixels

- Monochrome color (binary) requires 160 KBMonochrome color (binary) requires 160 KB

- High resolution color requires 5.2 MBHigh resolution color requires 5.2 MB

• Raster Displays (early 70s)Raster Displays (early 70s)

– like television, scan all pixels in regular patternlike television, scan all pixels in regular pattern

– use frame buffer (video RAM) to eliminate sync problemsuse frame buffer (video RAM) to eliminate sync problems

• RAMRAM

– ¼ MB (256 KB) cost $2 million in 1971¼ MB (256 KB) cost $2 million in 1971

– Do some math…Do some math…

- 1280 x 1024 screen resolution = 1,310,720 pixels1280 x 1024 screen resolution = 1,310,720 pixels

- Monochrome color (binary) requires 160 KBMonochrome color (binary) requires 160 KB

- High resolution color requires 5.2 MBHigh resolution color requires 5.2 MB

Display Technology: LCDs

Liquid Crystal Displays (LCDs)Liquid Crystal Displays (LCDs)

• LCDs: organic molecules, naturally in crystalline LCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E fieldstate, that liquefy when excited by heat or E field

• Crystalline state twists polarized light 90º. Crystalline state twists polarized light 90º.



• Crystalline state twists polarized light 90º. Crystalline state twists polarized light 90º.




• Crystalline state twists polarized light 90ºCrystalline state twists polarized light 90º



• Crystalline state twists polarized light 90ºCrystalline state twists polarized light 90º


Transmissive & reflective LCDs:Transmissive & reflective LCDs:

• LCDs act as light valves, not light emitters, and thus rely on an LCDs act as light valves, not light emitters, and thus rely on an external light source.external light source.

• Laptop screenLaptop screen

– backlitbacklit

– transmissive displaytransmissive display

• Palm Pilot/Game BoyPalm Pilot/Game Boy

– reflective displayreflective display

Transmissive & reflective LCDs:Transmissive & reflective LCDs:

• LCDs act as light valves, not light emitters, and thus rely on an LCDs act as light valves, not light emitters, and thus rely on an external light source.external light source.

• Laptop screenLaptop screen

– backlitbacklit

– transmissive displaytransmissive display

• Palm Pilot/Game BoyPalm Pilot/Game Boy

– reflective displayreflective display

Display Technology: Plasma

Plasma display panelsPlasma display panels

• Similar in principle to Similar in principle to fluorescent light tubesfluorescent light tubes

• Small gas-filled capsules Small gas-filled capsules are excited by electric field,are excited by electric field,emits UV lightemits UV light

• UV excites phosphorUV excites phosphor

• Phosphor relaxes, emits Phosphor relaxes, emits some other colorsome other color

Plasma display panelsPlasma display panels

• Similar in principle to Similar in principle to fluorescent light tubesfluorescent light tubes

• Small gas-filled capsules Small gas-filled capsules are excited by electric field,are excited by electric field,emits UV lightemits UV light

• UV excites phosphorUV excites phosphor

• Phosphor relaxes, emits Phosphor relaxes, emits some other colorsome other color

Display Technology

Plasma Display Panel ProsPlasma Display Panel Pros• Large viewing angleLarge viewing angle

• Good for large-format displaysGood for large-format displays

• Fairly brightFairly bright

ConsCons• ExpensiveExpensive

• Large pixels (~1 mm versus ~0.2 mm)Large pixels (~1 mm versus ~0.2 mm)

• Phosphors gradually depletePhosphors gradually deplete

• Less bright than CRTs, using more powerLess bright than CRTs, using more power

Plasma Display Panel ProsPlasma Display Panel Pros• Large viewing angleLarge viewing angle

• Good for large-format displaysGood for large-format displays

• Fairly brightFairly bright

ConsCons• ExpensiveExpensive

• Large pixels (~1 mm versus ~0.2 mm)Large pixels (~1 mm versus ~0.2 mm)

• Phosphors gradually depletePhosphors gradually deplete

• Less bright than CRTs, using more powerLess bright than CRTs, using more power

Display Technology: DMD / DLP

Digital Micromirror Devices (projectors) or Digital Micromirror Devices (projectors) or Digital Light ProcessingDigital Light Processing

• Microelectromechanical (MEM) devices, fabricated Microelectromechanical (MEM) devices, fabricated with VLSI techniqueswith VLSI techniques

Digital Micromirror Devices (projectors) or Digital Micromirror Devices (projectors) or Digital Light ProcessingDigital Light Processing

• Microelectromechanical (MEM) devices, fabricated Microelectromechanical (MEM) devices, fabricated with VLSI techniqueswith VLSI techniques

Display Technology: DMD / DLP

• DMDs are truly digital pixelsDMDs are truly digital pixels

• Vary grey levels by modulating pulse lengthVary grey levels by modulating pulse length

• Color: multiple chips, or color-wheelColor: multiple chips, or color-wheel

• Great resolutionGreat resolution

• Very brightVery bright

• Flicker problemsFlicker problems

• DMDs are truly digital pixelsDMDs are truly digital pixels

• Vary grey levels by modulating pulse lengthVary grey levels by modulating pulse length

• Color: multiple chips, or color-wheelColor: multiple chips, or color-wheel

• Great resolutionGreat resolution

• Very brightVery bright

• Flicker problemsFlicker problems

Display Technologies: Organic LED Arrays

Organic Light-Emitting Diode (OLED) ArraysOrganic Light-Emitting Diode (OLED) Arrays

• The display of the future? Many think so.The display of the future? Many think so.

• OLEDs function like regular semiconductor LEDsOLEDs function like regular semiconductor LEDs

• But they emit lightBut they emit light

– Thin-film deposition of organic, light-Thin-film deposition of organic, light-emitting molecules through vapor emitting molecules through vapor sublimation in a vacuum.sublimation in a vacuum.

– Dope emissive layers with fluorescent Dope emissive layers with fluorescent molecules to create color.molecules to create color.

Organic Light-Emitting Diode (OLED) ArraysOrganic Light-Emitting Diode (OLED) Arrays

• The display of the future? Many think so.The display of the future? Many think so.

• OLEDs function like regular semiconductor LEDsOLEDs function like regular semiconductor LEDs

• But they emit lightBut they emit light

– Thin-film deposition of organic, light-Thin-film deposition of organic, light-emitting molecules through vapor emitting molecules through vapor sublimation in a vacuum.sublimation in a vacuum.

– Dope emissive layers with fluorescent Dope emissive layers with fluorescent molecules to create color.molecules to create color.

http://www.kodak.com/global/en/professional/products/specialProducts/OEL/creating.jhtml


OLED pros:OLED pros:• TransparentTransparent

• FlexibleFlexible

• Light-emitting, and quite bright (daylight visible)Light-emitting, and quite bright (daylight visible)

• Large viewing angleLarge viewing angle

• Fast (< 1 microsecond off-on-off)Fast (< 1 microsecond off-on-off)

• Can be made large or smallCan be made large or small

• Available for cell phones and car stereosAvailable for cell phones and car stereos

OLED pros:OLED pros:• TransparentTransparent

• FlexibleFlexible

• Light-emitting, and quite bright (daylight visible)Light-emitting, and quite bright (daylight visible)

• Large viewing angleLarge viewing angle

• Fast (< 1 microsecond off-on-off)Fast (< 1 microsecond off-on-off)

• Can be made large or smallCan be made large or small

• Available for cell phones and car stereosAvailable for cell phones and car stereos


OLED cons:OLED cons:• Not very robust, display lifetime a key issueNot very robust, display lifetime a key issue

• Currently only passive matrix displaysCurrently only passive matrix displays

– Passive matrix:Passive matrix: Pixels are illuminated in scanline Pixels are illuminated in scanline order (like a raster display), but the lack of order (like a raster display), but the lack of phospherescence causes flickerphospherescence causes flicker

– Active matrix:Active matrix: A polysilicate layer provides thin film A polysilicate layer provides thin film transistors at each pixel, allowing direct pixel transistors at each pixel, allowing direct pixel access and constant illuminationaccess and constant illumination

See See http://www.howstuffworks.com/lcd4.htmhttp://www.howstuffworks.com/lcd4.htm for more info for more info

OLED cons:OLED cons:• Not very robust, display lifetime a key issueNot very robust, display lifetime a key issue

• Currently only passive matrix displaysCurrently only passive matrix displays

– Passive matrix:Passive matrix: Pixels are illuminated in scanline Pixels are illuminated in scanline order (like a raster display), but the lack of order (like a raster display), but the lack of phospherescence causes flickerphospherescence causes flicker

– Active matrix:Active matrix: A polysilicate layer provides thin film A polysilicate layer provides thin film transistors at each pixel, allowing direct pixel transistors at each pixel, allowing direct pixel access and constant illuminationaccess and constant illumination

See See http://www.howstuffworks.com/lcd4.htmhttp://www.howstuffworks.com/lcd4.htm for more info for more info

Movie Theaters

U.S. film projectors play film at 24 fpsU.S. film projectors play film at 24 fps• Projectors have a shutter to block light during frame advanceProjectors have a shutter to block light during frame advance

• To reduce flicker, shutter opens twice for each frame – resulting in 48 To reduce flicker, shutter opens twice for each frame – resulting in 48 fps flashingfps flashing

• 48 fps is perceptually acceptable48 fps is perceptually acceptable

European film projectors play film at 25 fpsEuropean film projectors play film at 25 fps• American films are played ‘as is’ in Europe, resulting in everything American films are played ‘as is’ in Europe, resulting in everything

moving 4% fastermoving 4% faster

• Faster movements and increased audio pitch are considered Faster movements and increased audio pitch are considered perceptually acceptableperceptually acceptable

U.S. film projectors play film at 24 fpsU.S. film projectors play film at 24 fps• Projectors have a shutter to block light during frame advanceProjectors have a shutter to block light during frame advance

• To reduce flicker, shutter opens twice for each frame – resulting in 48 To reduce flicker, shutter opens twice for each frame – resulting in 48 fps flashingfps flashing

• 48 fps is perceptually acceptable48 fps is perceptually acceptable

European film projectors play film at 25 fpsEuropean film projectors play film at 25 fps• American films are played ‘as is’ in Europe, resulting in everything American films are played ‘as is’ in Europe, resulting in everything

moving 4% fastermoving 4% faster

• Faster movements and increased audio pitch are considered Faster movements and increased audio pitch are considered perceptually acceptableperceptually acceptable

Viewing Movies at Home

Film to DVD transferFilm to DVD transfer

• Problem: 24 film fps must be converted to Problem: 24 film fps must be converted to

– NTSC U.S. television interlaced 29.97 fps 768x494 NTSC U.S. television interlaced 29.97 fps 768x494

– PAL Europe television 25 fps 752x582PAL Europe television 25 fps 752x582

Use 3:2 PulldownUse 3:2 Pulldown

• First frame of movie is broken into first three fields (odd, even, odd)First frame of movie is broken into first three fields (odd, even, odd)

• Next frame of movie is broken into next two fields (even, odd)Next frame of movie is broken into next two fields (even, odd)

• Next frame of movie is broken into next three fields (even, odd, even)…Next frame of movie is broken into next three fields (even, odd, even)…

Film to DVD transferFilm to DVD transfer

• Problem: 24 film fps must be converted to Problem: 24 film fps must be converted to

– NTSC U.S. television interlaced 29.97 fps 768x494 NTSC U.S. television interlaced 29.97 fps 768x494

– PAL Europe television 25 fps 752x582PAL Europe television 25 fps 752x582

Use 3:2 PulldownUse 3:2 Pulldown

• First frame of movie is broken into first three fields (odd, even, odd)First frame of movie is broken into first three fields (odd, even, odd)

• Next frame of movie is broken into next two fields (even, odd)Next frame of movie is broken into next two fields (even, odd)

• Next frame of movie is broken into next three fields (even, odd, even)…Next frame of movie is broken into next three fields (even, odd, even)…

Additional Displays

Display WallsDisplay Walls

• PrincetonPrinceton

• StanfordStanford

• UVa – Greg HumphreysUVa – Greg Humphreys

Display WallsDisplay Walls

• PrincetonPrinceton

• StanfordStanford

• UVa – Greg HumphreysUVa – Greg Humphreys

Display Wall Alignment

Additional Displays

StereoStereoStereoStereo

Visual System

We’ll discuss more fully later in semester but…We’ll discuss more fully later in semester but…

• Our eyes don’t mind smoothing across timeOur eyes don’t mind smoothing across time

– Still pictures appear to animateStill pictures appear to animate

• Our eyes don’t mind smoothing across spaceOur eyes don’t mind smoothing across space

– Discrete pixels blend into continuous color sheetsDiscrete pixels blend into continuous color sheets

We’ll discuss more fully later in semester but…We’ll discuss more fully later in semester but…

• Our eyes don’t mind smoothing across timeOur eyes don’t mind smoothing across time

– Still pictures appear to animateStill pictures appear to animate

• Our eyes don’t mind smoothing across spaceOur eyes don’t mind smoothing across space

– Discrete pixels blend into continuous color sheetsDiscrete pixels blend into continuous color sheets

Mathematical Foundations

Angel appendix B and CAngel appendix B and C

I’ll give a brief, informal review of some of the I’ll give a brief, informal review of some of the mathematical tools we’ll employmathematical tools we’ll employ• Geometry (2D, 3D)Geometry (2D, 3D)

• TrigonometryTrigonometry

• Vector spaces Vector spaces

– Points, vectors, and coordinatesPoints, vectors, and coordinates

• Dot and cross productsDot and cross products

Angel appendix B and CAngel appendix B and C

I’ll give a brief, informal review of some of the I’ll give a brief, informal review of some of the mathematical tools we’ll employmathematical tools we’ll employ• Geometry (2D, 3D)Geometry (2D, 3D)

• TrigonometryTrigonometry

• Vector spaces Vector spaces

– Points, vectors, and coordinatesPoints, vectors, and coordinates

• Dot and cross productsDot and cross products

Scalar Spaces

• Scalars: Scalars: … …

• Addition and multiplication (+ and Addition and multiplication (+ and ) operations defined) operations defined

• Scalar operations areScalar operations are

– Associative: Associative:

– Commutative: Commutative:

– Distributive: Distributive:

• Scalars: Scalars: … …

• Addition and multiplication (+ and Addition and multiplication (+ and ) operations defined) operations defined

• Scalar operations areScalar operations are

– Associative: Associative:

– Commutative: Commutative:

– Distributive: Distributive:

Scalar Spaces

• Additive Identity = 0Additive Identity = 0

• Multiplicative Identity = 1Multiplicative Identity = 1

• Additive Inverse = -Additive Inverse = -

• Multiplicative Inverse= Multiplicative Inverse= -1-1

• Additive Identity = 0Additive Identity = 0

• Multiplicative Identity = 1Multiplicative Identity = 1

• Additive Inverse = -Additive Inverse = -

• Multiplicative Inverse= Multiplicative Inverse= -1-1

Vector Spaces

Two types of elements:Two types of elements:

• Scalars (real numbers): Scalars (real numbers): … …

• Vectors (n-tuples):Vectors (n-tuples): uu, , vv, , ww, , … …

Operations:Operations:

• AdditionAddition

• SubtractionSubtraction

Two types of elements:Two types of elements:

• Scalars (real numbers): Scalars (real numbers): … …

• Vectors (n-tuples):Vectors (n-tuples): uu, , vv, , ww, , … …

Operations:Operations:

• AdditionAddition

• SubtractionSubtraction

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

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

• Add a new primitive, the pointAdd a new primitive, the point

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

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

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

• 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

• Add a new primitive, the pointAdd a new primitive, the point

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

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

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

Points

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

Coordinate Systems

Y

X

Z

Right-handedcoordinatesystemZ

X

Y

Left-handedcoordinatesystem

Grasp z-axis with handThumb points in direction of z-axis Roll fingers from positive x-axis towards positive y-axis

Euclidean Spaces

• Euclidean spaces permit the definition of distanceEuclidean spaces permit the definition of distance

• Dot product - distance between two vectorsDot product - distance between two vectors

• Projection of one vector onto anotherProjection of one vector onto another

• Euclidean spaces permit the definition of distanceEuclidean spaces permit the definition of distance

• Dot product - distance between two vectorsDot product - distance between two vectors

• Projection of one vector onto anotherProjection of one vector onto another

Euclidean Spaces

• We commonly use vectors to represent:We commonly use vectors to represent:

– Points in space (i.e., location)Points in space (i.e., location)

– Displacements from point to pointDisplacements from point to point

– Direction (i.e., orientation)Direction (i.e., orientation)

• We frequently use these operationsWe frequently use these operations

– Dot ProductDot Product

– Cross ProductCross Product

– NormNorm

• We commonly use vectors to represent:We commonly use vectors to represent:

– Points in space (i.e., location)Points in space (i.e., location)

– Displacements from point to pointDisplacements from point to point

– Direction (i.e., orientation)Direction (i.e., orientation)

• We frequently use these operationsWe frequently use these operations

– Dot ProductDot Product

– Cross ProductCross Product

– NormNorm

Scalar Multiplication

• Scalar multiplication:Scalar multiplication:

– Distributive rule:Distributive rule: ((uu + + vv) = ) = ((uu) + ) + ((vv))

(( + + ))uu = = uu + + uu

• Scalar multiplication “streches” a vector, changing its Scalar multiplication “streches” a vector, changing its length (length (magnitudemagnitude) but not its direction) but not its direction

• Scalar multiplication:Scalar multiplication:

– Distributive rule:Distributive rule: ((uu + + vv) = ) = ((uu) + ) + ((vv))

(( + + ))uu = = uu + + uu

• Scalar multiplication “streches” a vector, changing its Scalar multiplication “streches” a vector, changing its length (length (magnitudemagnitude) but not its direction) but not its direction

Dot Product

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

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

• 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

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

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

• 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

21 )(

y x y x

z x z x

z y z y

vv

Cross Product Right Hand Rule

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

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

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

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















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













2D Geometry

Know your high school geometry:Know your high school geometry:

• Total angle around a circle is 360° or 2π radiansTotal angle around a circle is 360° or 2π radians

• When two lines cross: When two lines cross:

– Opposite angles are equivalentOpposite angles are equivalent

– Angles along line sum to 180° Angles along line sum to 180°

• Similar triangles:Similar triangles:

– All corresponding angles are equivalentAll corresponding angles are equivalent

Know your high school geometry:Know your high school geometry:

• Total angle around a circle is 360° or 2π radiansTotal angle around a circle is 360° or 2π radians

• When two lines cross: When two lines cross:

– Opposite angles are equivalentOpposite angles are equivalent

– Angles along line sum to 180° Angles along line sum to 180°

• Similar triangles:Similar triangles:

– All corresponding angles are equivalentAll corresponding angles are equivalent

Trigonometry

Sine: “opposite over hypotenuse”Sine: “opposite over hypotenuse”

Cosine: “adjacent over hypotenuse”Cosine: “adjacent over hypotenuse”

Tangent: “opposite over adjacent”Tangent: “opposite over adjacent”

Unit circle definitions:Unit circle definitions:

• sin (sin () = x) = x

• cos (cos () = y) = y

• tan (tan () = x/y) = x/y

• etc…etc…

Sine: “opposite over hypotenuse”Sine: “opposite over hypotenuse”

Cosine: “adjacent over hypotenuse”Cosine: “adjacent over hypotenuse”

Tangent: “opposite over adjacent”Tangent: “opposite over adjacent”

Unit circle definitions:Unit circle definitions:

• sin (sin () = x) = x

• cos (cos () = y) = y

• tan (tan () = x/y) = x/y

• etc…etc…

(x, y)

Slope-intercept Line Equation

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)

Parametric Line Equation

Given points P1 = (x1, y1) and P2 = (x2, y2)Given points P1 = (x1, y1) and P2 = (x2, y2)x = x1 + t(x2 - x1)x = x1 + t(x2 - x1)y = y1 + t(y2 - y1)y = y1 + t(y2 - y1)

When:When:

• t=0, we get (x1, y1)t=0, we get (x1, y1)


• (0<t<1), we get points(0<t<1), we get pointson the segment betweenon the segment between(x1, y1) and (x2, y2)(x1, y1) and (x2, y2)

Given points P1 = (x1, y1) and P2 = (x2, y2)Given points P1 = (x1, y1) and P2 = (x2, y2)x = x1 + t(x2 - x1)x = x1 + t(x2 - x1)y = y1 + t(y2 - y1)y = y1 + t(y2 - y1)

When:When:



• (0<t<1), we get points(0<t<1), we get pointson the segment betweenon the segment between(x1, y1) and (x2, y2)(x1, y1) and (x2, y2) x

y

P2 = (x2, y2)

P1 = (x1, y1)

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

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

Reading

Chapters 1 and Appendix B of AngelChapters 1 and Appendix B of AngelChapters 1 and Appendix B of AngelChapters 1 and Appendix B of Angel

Documents

CRTs – A Review CRT technology hasn’t changed much in 50 yearsCRT technology hasn’t changed much in 50 years Early television technologyEarly television