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Modeling Transformation. Overview. 2D Transformation Basic 2D transformations Matrix representation Matrix Composition 3D Transformation Basic 3D transformations Same as 2D Transformation Hierarchies Scene graphs Ray casting. Modeling Transformation. - PowerPoint PPT Presentation

### Transcript of Modeling Transformation

• Modeling Transformation

• Overview2D TransformationBasic 2D transformationsMatrix representationMatrix Composition3D TransformationBasic 3D transformationsSame as 2DTransformation HierarchiesScene graphsRay casting

• Modeling TransformationSpecify transformations for objectsAllow definitions of objects in own coordinate systemsAllow use of object definition multiple times in a sceneRef] Hearn & Baker

• 2D Modeling Transformations

• 2D Modeling Transformations

• Basic 2D TransformationsTranslationx = x + txy = y + tyScalex = x * sxy = y * syShearx = x + hx * yy = y + hy * xRotationx = x * cosq - y * sinqy = x * sinq + y * cosq

• Matrix RepresentationRepresent 2D transformation by a matrix

Multiply matrix by column vectorApply transformation to point

• Matrix RepresentationTransformations combined by multiplication

Matrices are a convenient and efficient way to represent a sequence of transformations

• 2 x 2 MatricesWhat types of transformations can be represented with a 2 x 2 matrix?2D Identity

2D Scale around (0, 0)

• 2 x 2 MatricesWhat types of transformations can be represented with a 2x 2 matrix?2D Rotate around (0, 0)

2D Shear shx = 2

• 2 x 2 MatricesWhat types of transformations can be represented with a 2x 2 matrix?2D Mirror (=Reflection) over Y axis

2D Mirror over (0, 0)

• 2 x 2 MatricesWhat types of transformations can be represented with a 2x 2 matrix?2D Translation ?

NO!Only linear 2D transformation can be presented with a 2 x 2 matrix

• Linear TransformationsLinear transformations are combinations of ScaleRotationShearMirror (=Reflection)Properties of linear transformationsSatisfies: Origin maps to originLines map to linesParallel line remain parallelRatios are preservedClosed under composition

• 2D Translation2D Translation represented by a 3 x 3 matrixPoint represented with homogeneous coordinate

• Homogeneous CoordinatesAdd a 3rd coordinate to every 2D point(x, y, w) represents a point at location (x/w, y/w)(x, y, 0) represents a point at infinity(0, 0, 0) is not allowed

• Basic 2D TransformationsBasic 2D Transformations as 3 x 3 matrices

• Affine TransformationsAffine Transformations are combinations of Linear transformationTranslation

Properties of affine transformationsOrigin does not necessarily map to originLines map to linesParallel line remain parallelRatios are preservedClosed under composition

• Transformation : Rigid-Body / Euclidean Transforms Rigid Body Transformation (=rigid motion) A transform made up of only translation and rotationPreserve the shape of the objects that they act on Preserves distancesPreserves anglesTranslationRotationRigid / EuclideanIdentity

• Transformation : Similitudes / Similarity TransformsSimilarity TransformationsPreserves anglesTranslation, Rotation, Isotropic scaling TranslationRotationRigid / EuclideanSimilitudesIsotropic ScalingIdentity

• Transformation : Linear TransformationsTranslationRotationRigid / EuclideanLinearSimilitudesIsotropic ScalingIdentityScalingShearReflectionTranslation is not linear

• Transformation : Affine Transformationspreserves parallel linesTranslationRotationRigid / EuclideanLinearSimilitudesIsotropic ScalingScalingShearReflectionIdentityAffine

• Transformation : Projective Transformationspreserves linesTranslationRotationRigid / EuclideanLinearAffineProjectiveSimilitudesIsotropic ScalingScalingShearReflectionPerspectiveIdentity

• Projective TransformationsProjective Transformations Affine transformationProjective warps

Properties of projective transformationsOrigin does not necessarily map to originLines map to linesParallel line do not necessarily remain parallelRatios are not preserved (but cross-ratios are)Closed under composition

• Cross-ratioGiven any four points A, B, C, and D, taken in order, define their cross ratio as

Cross Ratio (ABCD) =cross ratio (ABCD) = cross ratio (A'B'C'D')

• Matrix CompositionTransformations can be combined by matrix multiplication

• Matrix CompositionMatrices are a convenient and efficient way to represent a sequence of transformationsGeneral purpose representationsHardware matrix multiplyEfficiency with premultiplication

• Matrix CompositionBe aware: order of transformations matterMatrix multiplication is not commutativeGlobalLocal

• Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS pTranslate then Scale: p' = S ( T p ) = ST p(0,0)(1,1)(4,2)(3,1)(8,4)(6,2)(0,0)(1,1)(2,2)(0,0)(5,3)(3,1)Scale(2,2)Translate(3,1)Translate(3,1)Scale(2,2)

• Matrix CompositionRotate by q around arbitrary point (a, b)M = T(a, b)* R(q) * T(-a, -b)Translate (a, b) to the origindo the rotation about origintranslate back

Scale by sx, sy around arbitrary point (a, b)M = T(a, b)* S(sx, sy) * T(-a, -b)Translate (a, b) to the origindo the scale about origintranslate back

• Transformation Game

• Overview2D TransformationBasic 2D transformationsMatrix representationMatrix Composition3D TransformationBasic 3D transformationsSame as 2DTransformation HierarchiesScene graphsRay casting

• 3D TransformationsSame idea as 2D transformationsHomogeneous coordinates: (x, y, z, w)4 x 4 transformation matrices

• Basic 3D transformations

• Basic 3D transformations

• Non-commutative Composition : Order of RotationsOrder of Rotation Affects Final PositionX-axis Z-axis

Z-axis X-axis

• Matrix Composition : Rotations in spaceRotation about an Arbitrary AxisRotate about some arbitrary a through angle Rotate about the z axis through an angle The axis a lies in the yz planeRotate about the x axis through an angle The axis a coincident with the z axisRotate about the a axis through Same as a rotation about the now coincident z axisReverse the second rotation about the x axisReverse the first rotation about the z axisReturning the axis a to its original position

yzxa

• Transformation HierarchiesScene may have hierarchy of coordinate systemsEach level stores matrix representing transformation from parents coordinate system

• Transformation Example 1 Well-suited for humanoid characters

• Transformation Example 1Ref] Princeton University

• Transformation Example 2An object may appear in a scene multiple times

• Transformation Example 2

• SummaryCoordinate systemsWorld coordinatesModeling coordinatesRepresentations of 3D modeling transformations4 x 4 MatricesScale, rotate, translate, shear, projections, etc.Not arbitrary warpsComposition of 3D transformationsMatrix multiplication (order matters)Transformation hierarchies