Blending Surfaces

Introduction Blending n. 1. The act of mingling 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 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 model We have seen a Belnding method before ! (where ?) Lets presents a simple scheme for point blending:

A General Blending Model Bezier and Bspline representation is exactly of this form. Q. Why use Points as the Base objects? A. There is no reason

A General Blending Model Let Q be an arbitrary parametrically defined objects. The general parametric equation is we receive is: Q – base objects b – blending functions

Blending example Blending 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 example We 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 Function We will use the Bernstein functions to create a smooth blending function. Let be the i-th Bernstein basis function of degree n. lets define :

Blending functions

General Equation Let S1 and S2 be two smooth surfaces then we can define:

Rail curves S is a blending surface smoothly connecting S1 and S2 along the rail curves S1(0,t), S2(1,t)

The intersection problem Finding 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 problem Solution: 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 Model Let c(t) be a smooth curve on [c,d] S1(s,t) a smooth surface on [a,b]X[c,d] We define:

Blending function

Curve / Surface Blending Model The new parametric surface we get is:

Curve / Surface Blending We 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 Blending C(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 S1 h(v) is a function from [0,1] to [a,b]X[c,d] For simplicity:

Curve / Surface Blending We need to create a blending erea. This is done by sweeping h(v) to the right. And the blending area is:

Curve / Surface Blending Thus the blending surface is:

3 surfaces – 2 curves Can we use a similar approach for more variables ? Yes we can …

Surface/Surface – Corner Blending

Blending is done in the parameter space. Intersection curve can be approximated !

Blending functions

Constructing b 1 definitions Bernstein of degree 5 f- mapping (rotation / translation)

Bernstein triangular C(s,t) = Bernstein triangular Edges are bizier curves. Fits our parameters (c1)

Blending function

Blend by pointwise interpolation Given 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 interpolation A general form of the vectors:

pointwise interpolation Using global functions M0 and M1 :

Blend by pointwise interpolation And the new surface is:

Choices of functions There are many choices for M and N. Tangent vectors T are more application driven. Example:

Geometric correspondence Hard problem There is No good solution.

Fanout surface technique Using intrinsic properties of the curves !

Fanout surface technique If P is a point on A. (the contact curve) And the curve becomes:

The fanout surface Using a and p as parameters gives us the fanout surface: And in a similar way:

Funout surfaces intersection The 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 solution

a=a(w), p=p(w), b=b(w), q=q(w) We have a parametric solution from degree 1 = curve !

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