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Identify the parts of a Bezier curve.
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A Bezier curve is defined by a few standard parts. Short list with what each part is and why it matters:
- Control points (P0, P1, …, Pn)
- The n+1 points that determine the shape. P0 and Pn are the curve endpoints; the interior control points pull the curve toward them but (for n>1) generally do not lie on the curve.
- Control polygon
- The polyline connecting the control points in order. It gives a visual guide to the curve shape and is used in algorithms (e.g., de Casteljau).
- Degree (n)
- The polynomial degree of the curve; a Bezier with n+1 control points is degree n. Degree controls flexibility/complexity.
- Parameter t
- A scalar parameter 0 ≤ t ≤ 1. The point on the curve is B(t). Moving t from 0 to 1 traces the curve.
- Basis (Bernstein) polynomials
- The blending functions b_{i,n}(t) = C(n,i) (1−t)^{n−i} t^i that weight the control points in the curve formula.
- Curve formula
- B(t) = Σ_{i=0..n} P_i · b_{i,n}(t). This is the explicit definition of the Bezier curve.
- End tangents / handles
- Tangent at start: B′(0) = n (P1 − P0). Tangent at end: B′(1) = n (Pn − P_{n−1}). The segments P0–P1 and P_{n−1}–Pn act as “handles.”
- de Casteljau construction
- A geometric, numerically stable recursive algorithm to evaluate B(t) and to split a Bezier curve into subcurves.
- Geometric properties
- Convex-hull property: the curve lies inside the convex hull of its control points.
- Variation diminishing: the curve does not oscillate more than the control polygon.
- Rational Bezier extension (if applicable)
- Adds non-negative weights w_i to control points; the curve is B(t) = (Σ w_i P_i b_{i,n}(t)) / (Σ w_i b_{i,n}(t)). Allows exact conic sections (circles, ellipses).
These are the main parts and concepts you’ll encounter when working with Bezier curves. If you want, I can illustrate with a diagram or a short numeric example.
- Control points (P0, P1, …, Pn)
- The n+1 points that determine the shape. P0 and Pn are the curve endpoints; the interior control points pull the curve toward them but (for n>1) generally do not lie on the curve.
- Control polygon
- The polyline connecting the control points in order. It gives a visual guide to the curve shape and is used in algorithms (e.g., de Casteljau).
- Degree (n)
- The polynomial degree of the curve; a Bezier with n+1 control points is degree n. Degree controls flexibility/complexity.
- Parameter t
- A scalar parameter 0 ≤ t ≤ 1. The point on the curve is B(t). Moving t from 0 to 1 traces the curve.
- Basis (Bernstein) polynomials
- The blending functions b_{i,n}(t) = C(n,i) (1−t)^{n−i} t^i that weight the control points in the curve formula.
- Curve formula
- B(t) = Σ_{i=0..n} P_i · b_{i,n}(t). This is the explicit definition of the Bezier curve.
- End tangents / handles
- Tangent at start: B′(0) = n (P1 − P0). Tangent at end: B′(1) = n (Pn − P_{n−1}). The segments P0–P1 and P_{n−1}–Pn act as “handles.”
- de Casteljau construction
- A geometric, numerically stable recursive algorithm to evaluate B(t) and to split a Bezier curve into subcurves.
- Geometric properties
- Convex-hull property: the curve lies inside the convex hull of its control points.
- Variation diminishing: the curve does not oscillate more than the control polygon.
- Rational Bezier extension (if applicable)
- Adds non-negative weights w_i to control points; the curve is B(t) = (Σ w_i P_i b_{i,n}(t)) / (Σ w_i b_{i,n}(t)). Allows exact conic sections (circles, ellipses).
These are the main parts and concepts you’ll encounter when working with Bezier curves. If you want, I can illustrate with a diagram or a short numeric example.
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