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

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Presentation on theme: "CRTs – A Review CRT technology hasn’t changed much in 50 yearsCRT technology hasn’t changed much in 50 years Early television technologyEarly television."— Presentation transcript:

1 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 –requires synchronization between video signal and electron beam vertical sync pulse Early computer displaysEarly computer displays –avoided synchronization using ‘vector’ algorithm –flicker 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 –requires synchronization between video signal and electron beam vertical sync pulse Early computer displaysEarly computer displays –avoided synchronization using ‘vector’ algorithm –flicker and refresh were problematic

2 CRTs – A Review Raster Displays (early 70s)Raster Displays (early 70s) –like television, scan all pixels in regular pattern –use frame buffer (video RAM) to eliminate sync problems RAMRAM –¼ MB (256 KB) cost $2 million in 1971 –Do some math… - 1280 x 1024 screen resolution = 1,310,720 pixels - Monochrome color (binary) requires 160 KB - High resolution color requires 5.2 MB Raster Displays (early 70s)Raster Displays (early 70s) –like television, scan all pixels in regular pattern –use frame buffer (video RAM) to eliminate sync problems RAMRAM –¼ MB (256 KB) cost $2 million in 1971 –Do some math… - 1280 x 1024 screen resolution = 1,310,720 pixels - Monochrome color (binary) requires 160 KB - High resolution color requires 5.2 MB

3 Display Technology: LCDs Liquid Crystal Displays (LCDs) LCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E fieldLCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E field Crystalline state twists polarized light 90º.Crystalline state twists polarized light 90º. Liquid Crystal Displays (LCDs) LCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E fieldLCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E field Crystalline state twists polarized light 90º.Crystalline state twists polarized light 90º.

4 Display Technology: LCDs Liquid Crystal Displays (LCDs) LCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E fieldLCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E field Crystalline state twists polarized light 90ºCrystalline state twists polarized light 90º Liquid Crystal Displays (LCDs) LCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E fieldLCDs: organic molecules, naturally in crystalline state, that liquefy when excited by heat or E field Crystalline state twists polarized light 90ºCrystalline state twists polarized light 90º

5 Display Technology: LCDs Transmissive & reflective LCDs: LCDs act as light valves, not light emitters, and thus rely on an external light source.LCDs act as light valves, not light emitters, and thus rely on an external light source. Laptop screenLaptop screen –backlit –transmissive display Palm Pilot/Game BoyPalm Pilot/Game Boy –reflective display Transmissive & reflective LCDs: LCDs act as light valves, not light emitters, and thus rely on an external light source.LCDs act as light valves, not light emitters, and thus rely on an external light source. Laptop screenLaptop screen –backlit –transmissive display Palm Pilot/Game BoyPalm Pilot/Game Boy –reflective display

6 Display Technology: Plasma Plasma display panels Similar in principle to fluorescent light tubesSimilar in principle to fluorescent light tubes Small gas-filled capsules are excited by electric field, emits UV lightSmall gas-filled capsules are excited by electric field, emits UV light UV excites phosphorUV excites phosphor Phosphor relaxes, emits some other colorPhosphor relaxes, emits some other color Plasma display panels Similar in principle to fluorescent light tubesSimilar in principle to fluorescent light tubes Small gas-filled capsules are excited by electric field, emits UV lightSmall gas-filled capsules are excited by electric field, emits UV light UV excites phosphorUV excites phosphor Phosphor relaxes, emits some other colorPhosphor relaxes, emits some other color

7 Display Technology Plasma Display Panel Pros Large viewing angleLarge viewing angle Good for large-format displaysGood for large-format displays Fairly brightFairly brightCons 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 Pros Large viewing angleLarge viewing angle Good for large-format displaysGood for large-format displays Fairly brightFairly brightCons 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

8 Display Technology: DMD / DLP Digital Micromirror Devices (projectors) or Digital Light Processing Microelectromechanical (MEM) devices, fabricated with VLSI techniquesMicroelectromechanical (MEM) devices, fabricated with VLSI techniques Digital Micromirror Devices (projectors) or Digital Light Processing Microelectromechanical (MEM) devices, fabricated with VLSI techniquesMicroelectromechanical (MEM) devices, fabricated with VLSI techniques

9 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

10 Display Technologies: Organic LED Arrays Organic 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- emitting molecules through vapor sublimation in a vacuum. –Dope emissive layers with fluorescent molecules to create color. Organic 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- emitting molecules through vapor sublimation in a vacuum. –Dope emissive layers with fluorescent molecules to create color. http://www.kodak.com/global/en/professional/products/specialProducts/OEL/creating.jhtml

11 Display Technologies: Organic LED Arrays 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: 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

12 Display Technologies: Organic LED Arrays 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: Pixels are illuminated in scanline order (like a raster display), but the lack of phospherescence causes flicker –Active matrix: A polysilicate layer provides thin film transistors at each pixel, allowing direct pixel access and constant illumination See http://www.howstuffworks.com/lcd4.htm for more info 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: Pixels are illuminated in scanline order (like a raster display), but the lack of phospherescence causes flicker –Active matrix: A polysilicate layer provides thin film transistors at each pixel, allowing direct pixel access and constant illumination See http://www.howstuffworks.com/lcd4.htm for more info

13 Movie Theaters U.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 fps flashingTo reduce flicker, shutter opens twice for each frame – resulting in 48 fps flashing 48 fps is perceptually acceptable48 fps is perceptually acceptable European film projectors play film at 25 fps American films are played ‘as is’ in Europe, resulting in everything moving 4% fasterAmerican films are played ‘as is’ in Europe, resulting in everything moving 4% faster Faster movements and increased audio pitch are considered perceptually acceptableFaster movements and increased audio pitch are considered perceptually acceptable U.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 fps flashingTo reduce flicker, shutter opens twice for each frame – resulting in 48 fps flashing 48 fps is perceptually acceptable48 fps is perceptually acceptable European film projectors play film at 25 fps American films are played ‘as is’ in Europe, resulting in everything moving 4% fasterAmerican films are played ‘as is’ in Europe, resulting in everything moving 4% faster Faster movements and increased audio pitch are considered perceptually acceptableFaster movements and increased audio pitch are considered perceptually acceptable

14 Viewing Movies at Home Film to DVD transfer Problem: 24 film fps must be converted toProblem: 24 film fps must be converted to –NTSC U.S. television interlaced 29.97 fps 768x494 –PAL Europe television 25 fps 752x582 Use 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 transfer Problem: 24 film fps must be converted toProblem: 24 film fps must be converted to –NTSC U.S. television interlaced 29.97 fps 768x494 –PAL Europe television 25 fps 752x582 Use 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)…

15

16 Additional Displays Display Walls PrincetonPrinceton StanfordStanford UVa – Greg HumphreysUVa – Greg Humphreys Display Walls PrincetonPrinceton StanfordStanford UVa – Greg HumphreysUVa – Greg Humphreys

17 Display Wall Alignment

18 Additional Displays StereoStereo

19 Visual System 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 animate Our eyes don’t mind smoothing across spaceOur eyes don’t mind smoothing across space –Discrete pixels blend into continuous color sheets 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 animate Our eyes don’t mind smoothing across spaceOur eyes don’t mind smoothing across space –Discrete pixels blend into continuous color sheets

20 Mathematical Foundations Angel appendix B and C I’ll give a brief, informal review of some of the mathematical tools we’ll employ Geometry (2D, 3D)Geometry (2D, 3D) TrigonometryTrigonometry Vector spacesVector spaces –Points, vectors, and coordinates Dot and cross productsDot and cross products Angel appendix B and C I’ll give a brief, informal review of some of the mathematical tools we’ll employ Geometry (2D, 3D)Geometry (2D, 3D) TrigonometryTrigonometry Vector spacesVector spaces –Points, vectors, and coordinates Dot and cross productsDot and cross products

21 Scalar Spaces Scalars:  …Scalars:  … Addition and multiplication (+ and  ) operations definedAddition and multiplication (+ and  ) operations defined Scalar operations areScalar operations are –Associative:  –Commutative:      –Distributive:                 Scalars:  …Scalars:  … Addition and multiplication (+ and  ) operations definedAddition and multiplication (+ and  ) operations defined Scalar operations areScalar operations are –Associative:  –Commutative:      –Distributive:                

22 Scalar Spaces Additive Identity = 0Additive Identity = 0 –  Multiplicative Identity = 1Multiplicative Identity = 1 –      Additive Inverse = - Additive Inverse = -  –  Multiplicative Inverse=  -1Multiplicative Inverse=  -1 –      Additive Identity = 0Additive Identity = 0 –  Multiplicative Identity = 1Multiplicative Identity = 1 –      Additive Inverse = - Additive Inverse = -  –  Multiplicative Inverse=  -1Multiplicative Inverse=  -1 –     

23 Vector Spaces Two types of elements: Scalars (real numbers):  …Scalars (real numbers):  … Vectors (n-tuples): u, v, w, …Vectors (n-tuples): u, v, w, …Operations: AdditionAddition SubtractionSubtraction Two types of elements: Scalars (real numbers):  …Scalars (real numbers):  … Vectors (n-tuples): u, v, w, …Vectors (n-tuples): u, v, w, …Operations: AdditionAddition SubtractionSubtraction

24 Vector Addition/Subtraction operation u + v, with:operation u + v, with: –Identity 0 v + 0 = v –Inverse - v + (- v ) = 0 Addition uses the “parallelogram rule”:Addition uses the “parallelogram rule”: operation u + v, with:operation u + v, with: –Identity 0 v + 0 = v –Inverse - v + (- v ) = 0 Addition uses the “parallelogram rule”:Addition uses the “parallelogram rule”: u+v u v u-v u v -v

25 Affine Spaces Vector spaces lack position and distanceVector spaces lack position and distance –They 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 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 location Add a new primitive, the pointAdd a new primitive, the point –Permits 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

26 Points Points support these operations Point-point subtraction: Q - P = vPoint-point subtraction: Q - P = v –Result is a vector pointing from P to Q Vector-point addition: P + v = QVector-point addition: P + v = Q –Result 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 operations Point-point subtraction: Q - P = vPoint-point subtraction: Q - P = v –Result is a vector pointing from P to Q Vector-point addition: P + v = QVector-point addition: P + v = Q –Result 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

27 Coordinate Systems Y X Z Right-handed coordinate system Z X Y Left-handed coordinate system l Grasp z-axis with hand l Thumb points in direction of z-axis l Roll fingers from positive x-axis towards positive y-axis

28 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

29 Euclidean Spaces We commonly use vectors to represent:We commonly use vectors to represent: –Points in space (i.e., location) –Displacements from point to point –Direction (i.e., orientation) We frequently use these operationsWe frequently use these operations –Dot Product –Cross Product –Norm We commonly use vectors to represent:We commonly use vectors to represent: –Points in space (i.e., location) –Displacements from point to point –Direction (i.e., orientation) We frequently use these operationsWe frequently use these operations –Dot Product –Cross Product –Norm

30 Scalar Multiplication Scalar multiplication:Scalar multiplication: –Distributive rule:  ( u + v ) =  ( u ) +  ( v ) (  +  ) u =  u +  u (  +  ) u =  u +  u Scalar multiplication “streches” a vector, changing its length (magnitude) but not its directionScalar multiplication “streches” a vector, changing its length (magnitude) but not its direction Scalar multiplication:Scalar multiplication: –Distributive rule:  ( u + v ) =  ( u ) +  ( v ) (  +  ) u =  u +  u (  +  ) u =  u +  u Scalar multiplication “streches” a vector, changing its length (magnitude) but not its directionScalar multiplication “streches” a vector, changing its length (magnitude) but not its direction

31 Dot Product The dot product or, more generally, inner product of two vectors is a scalar:The dot product or, more generally, inner product of two vectors is a scalar: v 1 v 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 (in 3D) Useful for many purposesUseful for many purposes Computing the length (Euclidean Norm) of a vector: length( v ) = ||v|| = sqrt( v v)Computing the length (Euclidean Norm) of a vector: length( v ) = ||v|| = sqrt( v v) Normalizing a vector, making it unit-length: v = v / ||v||Normalizing a vector, making it unit-length: v = v / ||v|| Computing the angle between two vectors:Computing the angle between two vectors: u v = |u| |v| cos(θ) Checking two vectors for orthogonalityChecking two vectors for orthogonality –u v = 0.0 The dot product or, more generally, inner product of two vectors is a scalar:The dot product or, more generally, inner product of two vectors is a scalar: v 1 v 2 = x 1 x 2 + y 1 y 2 + z 1 z 2 (in 3D) Useful for many purposesUseful for many purposes Computing the length (Euclidean Norm) of a vector: length( v ) = ||v|| = sqrt( v v)Computing the length (Euclidean Norm) of a vector: length( v ) = ||v|| = sqrt( v v) Normalizing a vector, making it unit-length: v = v / ||v||Normalizing a vector, making it unit-length: v = v / ||v|| Computing the angle between two vectors:Computing the angle between two vectors: u v = |u| |v| cos(θ) Checking two vectors for orthogonalityChecking two vectors for orthogonality –u v = 0.0 u θ v

32 Projecting one vector onto another If v is a unit vector and we have another vector, wIf v is a unit vector and we have another vector, w We can project w perpendicularly onto vWe can project w perpendicularly onto v And the result, u, has length w vAnd the result, u, has length w v Projecting one vector onto another If v is a unit vector and we have another vector, wIf v is a unit vector and we have another vector, w We can project w perpendicularly onto vWe can project w perpendicularly onto v And the result, u, has length w vAnd the result, u, has length w v Dot Product u w v

33 Is commutative u v = v uu v = v u Is distributive with respect to addition u (v + w) = u v + u wu (v + w) = u v + u w Is commutative u v = v uu v = v u Is distributive with respect to addition u (v + w) = u v + u wu (v + w) = u v + u w

34 Cross Product The cross product or vector product of two vectors is a vector: The cross product of two vectors is orthogonal to both Right-hand rule dictates direction of cross product The cross product or vector product of two vectors is a vector: The cross product of two vectors is orthogonal to both Right-hand rule dictates direction of cross product

35 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 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

36 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 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

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

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

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

40 2D 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 equivalent –Angles along line sum to 180° Similar triangles:Similar triangles: –All corresponding angles are equivalent 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 equivalent –Angles along line sum to 180° Similar triangles:Similar triangles: –All corresponding angles are equivalent

41 Trigonometry Sine: “opposite over hypotenuse” Cosine: “adjacent over hypotenuse” Tangent: “opposite over adjacent” Unit circle definitions: sin (  ) = xsin (  ) = x cos (  ) = ycos (  ) = y tan (  ) = x/ytan (  ) = x/y etc…etc… Sine: “opposite over hypotenuse” Cosine: “adjacent over hypotenuse” Tangent: “opposite over adjacent” Unit circle definitions: sin (  ) = xsin (  ) = x cos (  ) = ycos (  ) = y tan (  ) = x/ytan (  ) = x/y etc…etc… (x, y)

42 Slope-intercept Line Equation Slope =m = rise / run Slope = (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1) Solve for y: y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1 or: y = mx + b Slope =m = rise / run Slope = (y - y1) / (x - x1) = (y2 - y1) / (x2 - x1) Solve for y: y = [(y2 - y1)/(x2 - x1)]x + [-(y2-y1)/(x2 - x1)]x1 + y1 or: y = mx + b x y P2 = (x2, y2) P1 = (x1, y1) P = (x, y)

43 Parametric Line Equation Given points P1 = (x1, y1) and P2 = (x2, y2) x = x1 + t(x2 - x1) y = y1 + t(y2 - y1) When: t=0, we get (x1, y1)t=0, we get (x1, y1) t=1, we get (x2, y2)t=1, we get (x2, y2) (0 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/11/3269221/slides/slide_43.jpg", "name": "Parametric Line Equation Given points P1 = (x1, y1) and P2 = (x2, y2) x = x1 + t(x2 - x1) y = y1 + t(y2 - y1) When: t=0, we get (x1, y1)t=0, we get (x1, y1) t=1, we get (x2, y2)t=1, we get (x2, y2) (0

44 Other helpful formulas Length = sqrt (x2 - x1) 2 + (y2 - y1) 2 Midpoint, p2, between p1 and p3 p2 = ((x1 + x3) / 2, (y1 + y3) / 2))p2 = ((x1 + x3) / 2, (y1 + y3) / 2)) 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) 2 + (y2 - y1) 2 Midpoint, p2, between p1 and p3 p2 = ((x1 + x3) / 2, (y1 + y3) / 2))p2 = ((x1 + x3) / 2, (y1 + y3) / 2)) 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

45 Reading Chapters 1 and Appendix B of Angel


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