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Berkeley COMPSCI 294 - Lecture Notes

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CS 294-13Advanced Computer GraphicsDifferential Geometry BasicsJames F. O’BrienAssociate ProfessorU.C. Berkeley1Monday, October 26, 2009Topics•Vector and Tensor Fields•Divergence, curl, etc. •Parametric Curves•Tangents, curvature, and etc.•Parametric Surfaces•Normals, tangents, curvature, etc.•Implicit Surfaces•Normals, curvature, etc.2Monday, October 26, 2009Vectors•A vectors defines a magnitude and direction•Not just a list of numbers•Particular numbers are an artifact of the coordinate system we chose•Not all coordinate systems are orthonormal•Nearly everything that is useful can be defined w/o coordinate system•Vectors transform like vectors•No set location (e.g. no root)•But may be functions of location3v�= A · vˆv = v/||v||||v||v =[vx,vy,vz]v = v(u)v = v(x, y)Monday, October 26, 2009Tensors•Tensors transform like tensors•Tensors used to define oriented quantities•Independent of coordinate system•Specific realization will depend on coordinate system•Cartesian tensors -- orthonormal coordinate system•General tensors -- non-orthonormal coordinate system•Tensors have rank •Not related to dimension of space•Rank 0 → scalars•Rank 1 → vectors•Rank 2 → matrices •Rank 3 → don’t work well in matrix-vector notation4e.g.T�= A · T · ATT�= A · T · A−1Monday, October 26, 2009•ExamplesTensors5a · b = aT· b = ||a|| ||b||Cos(∠ab)a · bT= P(A · a) · (A · b)T=A · (a · bT) · AT= A · P · AT⇢�1√21√2−1√21√2�·�1002�·�1√2−1√21√21√2�=�−1232−3212�v1v2v1· vT1+ v2· vT2= SNote the way inner and otter products behave...R = x�· xT+ y�· yT+ z�· zTMonday, October 26, 2009•Notation due to Einstein•Makes life much easier•Takes a while to get used to•Useful in other contexts as well•Free index •Appears on both sides •Unique in each term•Implied “for all”•Dummy index•Appears exactly twice in each term•Implied “sum over”•Different for general tensorsSummation Notation6a −→ aiA −→ Aijc = A · b −→ ci= AijbjA�= RART−→ A�ij= AklRikRjls = a · b −→ s = aibiA = a · bT−→ Aij= aibjcT= bT· AT−→ ci= bjAijc = A · b −→ ci= bjAijMonday, October 26, 2009•Two special symbols•Delta•PermutationSummation Notation7δijδij=�1 if i = j0 if i �= jεijkεijk=1 if i, j, k are even permutation of 1, 2, 3−1 if i, j, k are odd permutation of 1, 2, 30 elseεij=1 if i, j are 1, 2−1 if i, j are 2, 10 elseIf you’re slumming in �2aiδij= ajεkijεkab= δiaδjb− δibδjaMonday, October 26, 2009Scalar Fields•Scalar as function of some spatial variable(s)•e.g.: 8Density Plot Height-field Plotf(x, y)=f(x) = Sin(x)Sin(y)Monday, October 26, 2009Vector Fields•Vector as function of some spatial variable(s)•e.g.: 9v(x, y)=v(x) = [Sin(x), Cos(y)]v(x, y)=v(x) = [1, Sin(y), 0]Monday, October 26, 2009Differential Operators on Fields•Derivatives of field w.r.t. spatial coordinates•Coordinates implicit given field parameterization•Linear operators on the field•Not tied to any particular coordinate system•Basic operators•Gradient•Divergence•Curl•Laplacian•All expressed with (a.k.a. Nabla or del)10∇ =[∂∂x,∂∂y,∂∂z]∇i= ∂i=∂∂xi∇∇ =�iei∂∂xiMonday, October 26, 2009Gradient•Often applied to scalar fields•Gives direction of steepest accent•Also has meaning for higher rank fields•Elevates rank by one•e.g. velocity gradient of a Newtonian fluid gives the strain rate11gradf(x)=∇f(x)=�∂f(x)∂x1,∂f(x)∂x2,∂f(x)∂x3,�f(x)=x2+ y2∇f(x) = [2x, 2y]Monday, October 26, 2009Divergence•For a vector field it describes the net expansion or contraction•Lowers rank by one•Divergence of vector field is a scalar•An inner product of derivatives with the field12∇ · [Sin(x), Cos(y)] = −Cos(x) + Sin(y)div v(x)=∇ · v(x)=∇T· v(x)=∂vx(x)∂x1+∂vy(x)∂x2+∂vz(x)∂x3Monday, October 26, 2009Curl•For a vector field it describes the net “rotation”•Cross product of derivatives with the field•Scaler in 2D, vector in 3D13curl v(x)=∇ × v(x)∇ × [Cos(y), 0] = −Sin(y)Monday, October 26, 2009Laplacian•Divergence of Gradient•Scalar second derivative operator•Difference between a point and its surround•Often used for smoothing of some sort14∇ · ∇ = ∇2=∂2∂xx+∂2∂yy+∂2∂zzcos2(x) sin2(y)2 cos2(x) cos2(y) − 4 cos2(x) sin2(y) + 2 sin2(x) sin2(y)Monday, October 26, 2009Notation Examples15v(x)=∇f(x) −→ vi= ∂ifs(x)=∇ · v(x) −→ s = ∂ivic(x)=∇ × v(x) −→ ci= εijk∂jvka(x)=(v(x) · ∇)b(x) −→ ai= vj∂jbiMonday, October 26, 2009Fun Facts•Helmholtz-Hodge decomposition•Smooth, differentiable vector field 16a = ∇s + ∇ × v + h∇s∇ × vhsolenoidal or divergence-free partirrotational or curl-free partharmonic part∇ · (∇ × v)=0Both are obvious in tensor notationScalar and vector potentials∇ × (∇s)=0Monday, October 26, 2009Directional Derivative17dfdx= x · ∇fAdd a picture or something...Monday, October 26, 2009•Curve is a geometric entity•Set of points in space•In neighborhood of any point it is isomorphic to a line•Generator function: •A vector valued function (careful with “vector”)•A scalar function for each dimension of embedding space•A particular parameterization is arbitrary and not unique•Parameterization is not intrinsicParametric Curves18�1.0�0.50.51.0�1.0�0.50.51.0�1.0�0.50.51.0�1.0�0.50.51.0[cos(θ), sin(θ)]�2uu2+1,1 − u2u2+1�x = x(t)Monday, October 26, 2009Derivatives•Given function for curve we can take derivatives w.r.t. the parameter:•The derivatives have names based on physical analogs•Velocity•Acceleration•Jerk•Snap, Crackle, and Pop•Speed is the magnitude of velocity•All are dependent on parameterization and not intrinsic•Note that, e.g., velocity is a vector field on 19˙x =dxdts = ||˙x||tMonday, October 26, 2009Arclength•Let• is the arclength of the curve •The arclength reparameterization of the curve is •The arclength parameterization is unique up to sign change and translation• and 20s = A(t)=�t0||x(τ )|| dτA(t)ˆx(s)=x(A−1(s))dˆx(s)ds=dx(t)dt��������dx(t)dt��������−1��������dˆx(s)ds��������=1Closed form arclength parameterization may be


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Berkeley COMPSCI 294 - Lecture Notes

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