ABSTRACT
In this chapter we study curves which are linear combinations of B-spline basis functions. We shall derive the properties of these curves. In addition, we develop the property which underlies most subdivision curve and surface algorithms as a property of uniform B-spline curves. We shall use the notation of the previous chapter and briefly review the most important definitions. Let https://www.w3.org/1998/Math/MathML"> u = u i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq3263.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denote a collection of distinct values, and m= https://www.w3.org/1998/Math/MathML"> m i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq3264.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> denote a set of positive integer values, one for each element of u. Define a nondecreasing sequence of real numbers https://www.w3.org/1998/Math/MathML"> t = t j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq3265.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> such that https://www.w3.org/1998/Math/MathML"> m i = https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq3266.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> https://www.w3.org/1998/Math/MathML"> c a r d t j : t j = u i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq3267.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and https://www.w3.org/1998/Math/MathML"> t j ≤ t j + 1 . u https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429064289/fc4a4622-76a5-4c96-97e4-82b5015d6fd6/content/eq3268.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and t can be infinite or finite.
