Bezier Curves Explained: Linear, Quadratic, and Cubic

Continuity measures how smoothly curves connect at joins. C⁰ continuity requires position continuity (no jumps). C¹ continuity adds velocity continuity (first derivative), requiring mirrored tangent points. C² continuity adds acceleration continuity, progressively constraining control points and reducing local control. C³ continuity is impossible for cubic Béziers without losing all control. Geometric continuity (Gⁿ) focuses on shape rather than parameterization: G¹ requires tangent direction continuity (normalizing velocity), G² requires curvature continuity (osculating circles), and G³ requires curvature rate continuity. Regular curves (non-zero velocity) satisfy Cⁿ → Gⁿ, but cusps violate this. Reflection seams in industrial design require G² continuity.

When joining two parametric curve pieces, parametric continuity (C0, C1, C2) requires matching parameter values, first derivatives, and second derivatives respectively at the joining point, while geometric continuity (G0, G1, G2) requires matching endpoints, proportional tangent vectors, and proportional curvature vectors respectively, with geometric continuity being a visual requirement that does not necessarily imply parametric continuity.

Geometric continuity conditions deal with the shape of the curve rather than the parametric representation. Zero order geometric continuity (G⁰) is the same as zero order parametric continuity - the curves must have the same coordinate position at the boundary point. First order geometric continuity (G¹) means the first parametric derivatives are proportional to each other at the intersection point, not necessarily equal. The tangent vector direction must be the same, but the magnitude may differ based on a constant k. Second order geometric continuity (G²) means both first and second order parametric derivatives are proportional at the boundary point.

This section presents the formal conditions for joining cubic Bézier curves with different levels of smoothness. It distinguishes between C-continuity (parametric smoothness requiring continuous derivatives as functions of parameter t) and G-continuity (geometric smoothness requiring matching geometric properties regardless of parameterization). For C0 continuity, curves share one control point. For G1 continuity, edge vectors must be parallel. For C1 continuity, edge vectors must be identical. For G2 continuity, non-linear conditions on the first three control points apply. The section emphasizes that C1 implies G1 for regular curves, but G1 does not imply C1, highlighting the importance of understanding both perspectives in geometric modeling.

This section presents the mathematical criteria for joining curves smoothly. C₀ continuity requires matching endpoints (Q₁(1)=Q₂(0)), ensuring curves meet at a common location. C₁ continuity adds matching tangent vectors (Q₁'(1)=Q₂'(0)), producing visually smooth transitions. G₁ continuity relaxes this to require only matching tangent directions (Q₁'(1)=k×Q₂'(0)), allowing different magnitudes. These continuity conditions form the basis for constructing complex curves from simpler segments while maintaining desired smoothness properties.

NURBS (Non-Uniform Rational B-Splines) generalize Bezier curves by automatically handling piecewise construction and providing smooth connections between segments. They require specifying order (degree+1), degree, and knot sequences. Uniform knots produce evenly spaced parameters, while non-uniform sequences enable custom spacing. Repeating knots forces curves through specific points. NURBS combine the mathematical rigor needed for industrial applications with the flexibility of arbitrary control point arrangements, forming the backbone of professional CAD and animation software.

B-spline curves offer significant advantages over Bezier curves by allowing independent specification of curve order (k) from the number of control points, enabling local propagation where changing a control point affects only nearby segments rather than the entire curve. The curve is composed of n-k+2 segments with C^(k-2) continuity, where k=3 provides slope continuity and k=4 provides curvature continuity. NURBS (Non-Uniform Rational B-splines) extend B-splines by introducing weights to control points, enabling exact representation of conic sections like circles, ellipses, parabolas, and hyperbolas, which cannot be achieved with standard B-splines or Bezier curves. This mathematical flexibility makes NURBS the industry standard for representing curves and surfaces in computer-aided design.

B-splines of order k are piecewise polynomials of degree k-1 with C^(k-2) continuity. Knot vectors define the intervals where basis functions are non-zero, with uniform versions having equally spaced knots. NURBS extend B-splines by introducing non-uniform knot vectors and rational weights w_i, providing additional design freedom. These weights enable exact representation of conic sections and combine the strengths of B-splines and Bezier curves. NURBS have become the industry standard for CAD data exchange due to their versatility in representing complex shapes with precise mathematical control.

B-Splines improve upon Bezier curves by providing local control, where each control point affects only a portion of the curve. NURBS (Non-Uniform Rational B-Splines) extend B-Splines by incorporating rational weights, enabling exact representation of conic sections. These techniques are essential for CAD applications and high-quality 3D modeling, providing flexibility in representing complex shapes while maintaining computational efficiency.

Non-Uniform Rational B-Splines (NURBS) extend B-splines by allowing control of the priority or weight of each control point, making curves gravitate toward or away from specific points. They also support non-uniform knot intervals. NURBS are particularly powerful because they can represent conic sections exactly, including perfect circles, which is crucial for precise geometric modeling.

The flubber.js library provides an interpolate function that calculates intermediate paths between two SVG paths. The interpolator takes two path 'd' attributes as input and returns a new path at any given progress value between 0 and 1. A progress value of 0 returns the original path, 1 returns the target path, and values in between produce intermediate shapes. This allows developers to morph between any two SVG shapes by calculating the mathematical approximation of their transformation.

SVG (Scalable Vector Graphics) is an XML-based format ideal for automated design generation. Unlike PDF with its complex internal structures, SVG uses text-based tags describing shapes and paths. To use fonts, you must convert font glyphs to SVG path data using specialized libraries. This allows programmatic generation of vector graphics with any font, enabling automated design production.

This section covers the practical aspects of extracting mathematical equations from vector graphics. Vector graphics store drawing instructions rather than pixel data, enabling lossless scaling unlike raster formats. SVG files contain path data as strings of letters followed by coordinates: 'M' moves to a point, 'Q' draws quadratic Bézier curves, and 'L' draws lines. Each new instruction implicitly starts at the previous endpoint, creating compact but cryptic representations requiring parsing. Quadratic Bézier curves are defined by three points (initial, control, final) with parametric equation B(t) = (1-t)²P₀ + 2(1-t)tP₁ + t²P₂. Programming implementation involves importing raw SVG text, parsing instructions, explicitly recording starting points, translating coordinates to center shapes, inverting y-axis, and pairing x/y coordinates for proper data structure organization.

NV_path_rendering is an OpenGL extension that implements GPU-accelerated path rendering using a stenciled in-cover (STC) approach, where paths are first stenciled into the stencil buffer and then covered with conservative geometry, enabling efficient rendering of resolution-independent 2D vector graphics with support for all major path rendering standards including dashing, join styles, text, and font rendering, while allowing integration with traditional 3D rendering pipelines.

CMDCam is a Minecraft mod that enables precise camera control for video production by allowing users to define paths that the camera follows, with different interpolation methods (linear, hermite, cubic, cosine) determining how smoothly the camera moves between points, and target settings (direct or indirect) controlling what the camera focuses on during the animation.