CS 445 / 645: Introductory Computer GraphicsPowerPoint PresentationSlide 3Slide 4Translation Matrices?Homogeneous CoordinatesSlide 7Slide 8Slide 9Slide 10Slide 11Translation MatricesSlide 13Slide 14Slide 15Transformation CommutativityMore On Homogeneous CoordsHomogenous CoordinatesProjection MatrixOrthographic ProjectionOrthographic: Screen Space TransformationSlide 22Perspective TransformationPerspective ProjectionSlide 25Slide 26Slide 27A Perspective Projection MatrixSlide 29Slide 30Projection MatricesA 3D SceneViewing Transformations2 Basic StepsSlide 35Creating Camera Coordinate SpaceConstructing Viewing Transformation, VSlide 38Slide 39Compositing Vectors to Form VSlide 41Slide 42Slide 43Final Viewing Transformation, VCS 445 / 645: CS 445 / 645: Introductory Computer Introductory Computer GraphicsGraphicsGeometric TransformsMatrix Manipulation - ExampleMatrix Manipulation - Example•Drawing a car with wheels and lugnutsdraw_wheel( );for (j=0; j<5; j++) {glPushMatrix ();glRotatef(72.0*j, 0.0, 0.0, 1.0);glTranslatef (3.0, 0.0, 0.0);draw_bolt ( );glPopMatrix ( );Matrix Manipulation - ExampleMatrix Manipulation - Exampledraw_wheel( );for (j=0; j<5; j++) {glPushMatrix ();glRotatef(72.0*j, 0.0, 0.0, 1.0);glTranslatef (3.0, 0.0, 0.0);draw_bolt ( );glPopMatrix ( );Global – Bottom UpStartRotTransMatrix Manipulation - ExampleMatrix Manipulation - Exampledraw_wheel( );for (j=0; j<5; j++) {glPushMatrix ();glRotatef(72.0*j, 0.0, 0.0, 1.0);glTranslatef (3.0, 0.0, 0.0);draw_bolt ( );glPopMatrix ( );Local – Top DownStartRotTransTranslation Matrices?Translation Matrices?•We can composite scale matrices just as we did rotation matrices•But how to represent translation as a matrix?•Answer: with homogeneous coordinatesHomogeneous CoordinatesHomogeneous Coordinates•Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector(Note that typically w = 1 in object coordinates)wzyxwzwywxzyx1///),,(Homogeneous CoordinatesHomogeneous Coordinates•Homogeneous coordinates seem unintuitive, but they make graphics operations much easier•Our transformation matrices are now 4x4:10000)cos()sin(00)sin()cos(00001xRHomogeneous CoordinatesHomogeneous Coordinates•Homogeneous coordinates seem unintuitive, but they make graphics operations much easier•Our transformation matrices are now 4x4:10000)cos(0)sin(00100)sin(0)cos(yRHomogeneous CoordinatesHomogeneous Coordinates•Homogeneous coordinates seem unintuitive, but they make graphics operations much easier•Our transformation matrices are now 4x4:1000010000)cos()sin(00)sin()cos(zRHomogeneous CoordinatesHomogeneous Coordinates•Homogeneous coordinates seem unintuitive, but they make graphics operations much easier•Our transformation matrices are now 4x4:1000000000000zyxSSSSHomogeneous CoordinatesHomogeneous Coordinates•How can we represent translation as a 4x4 matrix?•A: Using the rightmost column:1000000000000zyxTTTTTranslation MatricesTranslation Matrices•Now that we can represent translation as a matrix, we can composite it with other transformations•Ex: rotate 90° about X, then 10 units down Z:wzyxwzyx10000)90cos()90sin(00)90sin()90cos(0000110001010000100001''''Translation MatricesTranslation Matrices•Now that we can represent translation as a matrix, we can composite it with other transformations•Ex: rotate 90° about X, then 10 units down Z:wzyxwzyx100000100100000110001010000100001''''Translation MatricesTranslation Matrices•Now that we can represent translation as a matrix, we can composite it with other transformations•Ex: rotate 90° about X, then 10 units down Z:wzyxwzyx10001001001000001''''Translation MatricesTranslation Matrices•Now that we can represent translation as a matrix, we can composite it with other transformations•Ex: rotate 90° about X, then 10 units down Z:wyzxwzyx10''''Transformation CommutativityTransformation Commutativity•Is matrix multiplication, in general, commutative? Does AB = BA?•What about rotation, scaling, and translation matrices? –Does RxRy = RyRx?–Does RAS = SRA ?–Does RAT = TRA ?More On Homogeneous More On Homogeneous CoordsCoords•What effect does the following matrix have?•Conceptually, the fourth coordinate w is a bit like a scale factorwzyxwzyx10000010000100001''''Homogenous CoordinatesHomogenous Coordinates•In particular, increasing w makes things smaller•We think of homogenous coordinates as defining a projective space–Increasing w “getting further away”•Will come in handy for projection matricesProjection MatrixProjection Matrix•We talked about geometric transforms, focusing on modeling transforms–Ex: translation, rotation, scale, gluLookAt()–These are encapsulated in the OpenGL modelview matrix•Can also express projection as a matrix•Next few slides: representing orthographic and perspective projection with the projection matrixOrthographic ProjectionOrthographic Projection•Simple OrthographicTransformation•Original world units are preserved–Pixel units are preferredOrthographic: Screen Space Orthographic: Screen Space TransformationTransformationtop=20 mbottom=10 mleft =10 m right = 20 m(0, 0)(max pixx, max pixy)(width in pixels)(height in pixels)Orthographic: Screen Space Orthographic: Screen Space TransformationTransformation•left, right, top, bottom refer to the viewing frustum in modeling coordinates•width and height are in pixel units•This matrix scales and translates to accomplish the transition in unitsPerspective TransformationPerspective Transformation•First discovered
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