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Transcript of Blending Surfaces
IntroductionBlending n. 1. The act of mingling. 1913 Webster
2. (Paint.) The method of laying on different tints so that they may mingle together while wet, and shade into each other insensibly. --Weale. 1913 Webster
Introduction The process of mixing several base objects to form a new object.The process of providing smooth transition between intersecting surfaces or smooth connection between disjoint surfaces.
A General Blending modelWe have seen a Belnding method before !(where ?)
Lets presents a simple scheme for point blending:
A General Blending ModelBezier and Bspline representation is exactly of this form.Q. Why use Points as the Base objects?A. There is no reason
A General Blending ModelLet Q be an arbitrary parametrically defined objects.The general parametric equation is we receive is:
Q base objectsb blending functions
Blending exampleBlending a set of curves for example:We use a continues function b which satisfy the following conditions:
Then blending and , two parametric curves on the same domain is:
Blending exampleWe can immediately see that:S is a surface.S(0,t) is a curve. (which one ?)S(1,t) is a curve. (which one ?)
Q. Can we blend in this way surfaces ?A. Yes
Blending FunctionWe will use the Bernstein functions to create a smooth blending function.Let be the i-th Bernstein basis function of degree n. lets define :
General EquationLet S1 and S2 be two smooth surfaces then we can define:
Rail curvesS is a blending surface smoothly connecting S1 and S2 along the rail curves S1(0,t) , S2(1,t)
The intersection problemFinding the intersection curve between two surfaces is a Hard problem.Algebraic solutions complex , good for low dimensionality.Numerically solutions not accurate, loose parameterization.
The intersection problemSolution:Numerically find points on the intersection curve.Construct a curve C that interpolate the points.Locally change the surfaces so they pass through C.
Curve/Surface Blending ModelLet c(t) be a smooth curve on [c,d]S1(s,t) a smooth surface on [a,b]X[c,d]We define:
Curve / Surface Blending ModelThe new parametric surface we get is:
Curve / Surface BlendingWe can easily see that the interpolated curve pass through the new Surface.To finish the algorithm we will use the model presented earlier on our problem.
Curve / Surface BlendingC(t) is a curve defined on [a,b]S1(s,t) is a surface defined on [a,b]x[c,d]C1=S1(h(v)) a curve on S1h(v) is a function from [0,1] to [a,b]X[c,d]
Curve / Surface BlendingWe need to create a blending erea.This is done by sweeping h(v) to the right.
And the blending area is:
Curve / Surface BlendingThus the blending surface is:
3 surfaces 2 curvesCan we use a similar approach for more variables ?
Yes we can
Surface/Surface Corner Blending
Surface/Surface Corner BlendingBlending is done in the parameter space.
Intersection curve can be approximated !
Constructing b1 definitionsBernstein of degree 5
f- mapping (rotation / translation)
Bernstein triangularC(s,t) = Bernstein triangularEdges are bizier curves.Fits our parameters (c1)
Blend by pointwise interpolationGiven two surfaces P(u,v) , Q(s,t)Let A(w) , B(w) two respective contact curves:A(w)=P(u(w),v(w))B(w)=Q(s(w),t(w))
We pick two vectors in the tangent plane.
pointwise interpolationA general form of the vectors:
pointwise interpolationUsing global functions M0 and M1 :
Blend by pointwise interpolationAnd the new surface is:
Choices of functionsThere are many choices for M and N.Tangent vectors T are more application driven.Example:
Geometric correspondenceHard problemThere is No good solution.
Fanout surface techniqueUsing intrinsic properties of the curves !
Fanout surface techniqueIf P is a point on A. (the contact curve)
And the curve becomes:
The fanout surfaceUsing a and p as parameters gives us the fanout surface:
And in a similar way:
Funout surfaces intersectionThe intersection of the fanout surfaces gives us the needed correspondence. 3 equations , 4 unknowns, one parameter
Q. Where are the 3 equations?A. Next page
Correspondence solutiona=a(w) , p=p(w) , b=b(w) , q=q(w)We have a parametric solution from degree 1 = curve !