In this section, we use the generating functions for the Bernstein basis functions to derive a family of functional equations. Using these equations, we derive a collection of identities for the Bernstein basis functions.
3.1 Sums and alternating sums
From (3), we get the following functional equations:
(8)
and
(9)
Theorem 3.1 (Sum of the Bernstein basis functions)
Proof From (8), one finds that
(10)
By combining (2) and (10), we get
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Theorem 3.2 (Alternating sum of the Bernstein basis functions)
Proof By combining (9) and (10), we obtain
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 3.3 Goldman [6], [[5], Chapter 5, pp.299-306] derived the formula for the alternating sum of the Bernstein basis functions algebraically.
3.2 Subdivision
From (3), we have the following functional equation:
(11)
From this functional equation, we get the following identity which is the basis for subdivision of Bezier curves, cf. [4–6, 15].
Theorem 3.4
Proof By equations (3) and (11),
Therefore
Substituting (1) into the above equation, we arrive at the desired result. □
Remark 3.5 Theorem 3.4 is a bit tricky to prove with algebraic manipulations. Goldman [6], [[5], Chapter 5, pp.299-306] proved this identity algebraically. He also proved the following related subdivision identities:
and
For additional identities, see [6], [[5], Chapter 5, pp.299-306].
3.3 Formula for the monomials in terms of the Bernstein basis functions
Multiplying both sides of (3) by , we get
Summing both sides of the above equation over k, we obtain the following functional equation, which is used to derive a formula for the monomials in terms of the Bernstein basis functions:
(12)
Theorem 3.6
Proof Combining (2) and (12), we get
Therefore
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
3.4 Differentiating the Bernstein basis functions
In this section we give higher-order derivatives of the Bernstein basis functions. We begin by observing that
(13)
where
and
Using Leibnitz’s formula for the l th derivative, with respect to x, we obtain the following higher-order partial differential equation:
(14)
From this equation, we arrive at the following theorem.
Theorem 3.7
(15)
Proof Formula (15) follows immediately from (14). □
Applying Theorem 3.7, we obtain a new derivation for the higher-order derivatives of the Bernstein basis functions.
Theorem 3.8
(16)
Proof By substituting the right-hand side of (2) into (15), we get
Therefore
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Substituting into (16), we arrive at the following standard corollary.
Corollary 3.9
cf. [1–16].
3.5 Recurrence relation
In the previous section we computed the derivative of (13) with respect to x to derive a derivative formula for the Bernstein basis functions. In this section we are going to differentiate (13) with respect to t to derive a recurrence relation for the Bernstein basis functions.
Using Leibnitz’s formula for the v th derivative, with respect to t, we obtain the following higher-order partial differential equation:
(17)
From the above equation, we have the following theorem.
Theorem 3.10
(18)
Proof Formula (18) follows immediately from (17). □
Using definitions (3) and (1) in Theorem 3.10, we obtain a recurrence relation for the Bernstein basis functions.
Theorem 3.11
(19)
Proof By substituting the right-hand side of (2) into (18), we get
Therefore
From the above equation, we get
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
We also computed the derivative of (3) with respect to t to derive the following higher-order partial differential equation:
By using the above equation, we derive another recurrence relation for the Bernstein basis functions as follows:
Remark 3.12 Setting in (19), one obtains the standard recurrence
3.6 Degree raising
In this section we present a functional equation which we apply to provide a new proof of the degree raising formula for the Bernstein polynomials.
From (3), we obtain the following functional equation:
Therefore
(20)
Substituting into the above equation, we have
(21)
The above relation can also be proved by (1), cf. [4–6].
From (3), we also get the following functional equation:
Therefore
Substituting , we have
(22)
Adding (21) and (22), we get the standard degree elevation formula