ABSTRACT
It is often reiterated that the local properties of B-splines make them very useful for representing geometry. The ability to support general, stable, rapid refinement algorithms help make them supportive of hierarchical design. For instance, in Figure 17.1 (a), the gross shape of the curve with the shown knot vector is desirable; however, some fine detail is lacking. In Figure 17.1(b), the knot vector has been supplemented generating a new control polygon for the same curve. Finally in Figure 17.2 the curve is truly modified. The flexibility introduced by the pseudoknots has been used, and those knots are real. Note that the modified curve is identical to the original in its gross shape, but has just the desired local perturbations. We present a general algorithm which implements Algorithm 16.9. (a) Initial curve and control polygon, <inline-formula> <alternatives> <mml:math display="inline" xmlns:mml="<a href="https://www.w3.org/1998/Math/MathML" target="_blank">https://www.w3.org/1998/Math/MathML</a>"> <mml:mi mathvariant="bold-italic">τ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mn>0,0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0,0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1,2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2,2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>}</mml:mo> </mml:math> <inline-graphic xlink:href="<a href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq7463.tif" target="_blank">https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq7463.tif</a>" xmlns:xlink="<a href="https://www.w3.org/1999/xlink" target="_blank">https://www.w3.org/1999/xlink</a>"/> </alternatives> </inline-formula>; (b) refined control polygon, <inline-formula> <alternatives> <mml:math display="inline" xmlns:mml="<a href="https://www.w3.org/1998/Math/MathML" target="_blank">https://www.w3.org/1998/Math/MathML</a>"> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mn>0,0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0,0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>0.6,0.8,1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1.2,1.4,2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2,2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>}</mml:mo> </mml:math> <inline-graphic xlink:href="<a href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq7464.tif" target="_blank">https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq7464.tif</a>" xmlns:xlink="<a href="https://www.w3.org/1999/xlink" target="_blank">https://www.w3.org/1999/xlink</a>"/> </alternatives> </inline-formula>. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/fig17_1.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/> Modified control polygon and resulting curve. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/fig17_2.jpg" xmlns:xlink="https://www.w3.org/1999/xlink"/>
