Let x(t) and y(t) determine two quadratic Bézier curves, with control points { $p_{0} , p_{1} , p_{2}$ } and { $p_{2} , p_{3} , p_{4}$ }, respectively. The curves are joined at $p_{2} = x(1) = y(0)$ .

a. Suppose the combined curve has $G^1$ continuity (at $p_{2}$ ). What algebraic restriction does this condition impose on the control points? Express this restriction in geometric language.
b. Repeat part (a) for $C^1$ continuity.

Question

Let x(t) and y(t) determine two quadratic Bézier curves, with control points {[latex] p_{0} , p_{1} , p_{2} [/latex]} and {[latex] p_{2} , p_{3} , p_{4} [/latex]}, respectively. The curves are joined at [latex] p_{2} = x(1) = y(0) [/latex].

a. Suppose the combined curve has [latex]G^1[/latex] continuity (at [latex] p_{2} [/latex]). What algebraic restriction does this condition impose on the control points? Express this restriction in geometric language.
b. Repeat part (a) for [latex]C^1[/latex] continuity.

Accepted Answer

a. From Example 1, [latex] x^′(1) = 2(p_{2} - p_{1}) [/latex]. Also, using the control points for y(t) in place of w(t), Example 1 shows that [latex] y^′(0) = 2(p_{3} - p_{2}) [/latex]. G¹ continuity means that [latex] y^′(0) = kx^′(1) [/latex] for some positive constant k. Equivalently,

[latex] p_{3} - p_{2} = k(p_{2} - p_{1}) , ext{with} k \gt 0 [/latex] (3)

Geometrically, (3) implies that [latex] p_{2}[/latex] lies on the line segment from [latex] p_{1}[/latex] to [latex] p_{3}[/latex]. To prove this, let [latex] t = (k + 1)^{-1} [/latex], and note that 0 < t < 1. Solve for k to obtain [latex] k = (1 - t)/t [/latex]. When this expression is used for k in (3), a rearrangement shows that [latex] p_{2} = (1 - t)p_{1} + tp_{3} [/latex], which verifies the assertion about [latex] p_{2} [/latex].

b. C¹ continuity means that [latex] y^′(0) = x^′(1) [/latex]. Thus [latex] 2(p_{3} - p_{2}) = 2(p_{2} - p_{1}) [/latex], so [latex] p_{3} - p_{2} = p_{2} - p_{1} [/latex], and [latex] p_{2} = (p_{1} + p_{3})/2 [/latex]. Geometrically, [latex] p_{2} [/latex] is the midpoint of the line segment from [latex] p_{1} [/latex] to [latex] p_{3} [/latex]. See Figure 3.

Question 8.6.2: Let x(t) and y(t) determine two quadratic Bézier curves, wit...

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