06-16-2026, 01:59 AM
Curves and Surfaces
Bartosz Ciechanowski
Summary
The article “Curves and Surfaces” by Bartosz Ciechanowski is an interactive exploration of how complex geometric shapes can be constructed and understood from simple mathematical ideas. It begins with curves defined through control points and linear interpolation, showing how smooth shapes emerge from weighted combinations of points.
It then extends these ideas to surfaces by combining two independent interpolation parameters, revealing how 3D surfaces can be built from grids of curves and how their geometry is governed by weighting functions.
The article explains key concepts in differential geometry such as curvature, osculating circles, and how curvature affects reflections and physical behavior like motion on curved paths. It further develops the idea of surface curvature, including principal curvatures and Gaussian curvature, and shows how smooth transitions (curvature continuity) are essential for realistic modeling in graphics and engineering. Finally, it introduces subdivision surfaces (Catmull–Clark) as a practical method for generating smooth, complex shapes from simple polygon meshes, widely used in computer graphics and animation.
ARTICLE PAGE
Bartosz Ciechanowski
Summary
The article “Curves and Surfaces” by Bartosz Ciechanowski is an interactive exploration of how complex geometric shapes can be constructed and understood from simple mathematical ideas. It begins with curves defined through control points and linear interpolation, showing how smooth shapes emerge from weighted combinations of points.
It then extends these ideas to surfaces by combining two independent interpolation parameters, revealing how 3D surfaces can be built from grids of curves and how their geometry is governed by weighting functions.
The article explains key concepts in differential geometry such as curvature, osculating circles, and how curvature affects reflections and physical behavior like motion on curved paths. It further develops the idea of surface curvature, including principal curvatures and Gaussian curvature, and shows how smooth transitions (curvature continuity) are essential for realistic modeling in graphics and engineering. Finally, it introduces subdivision surfaces (Catmull–Clark) as a practical method for generating smooth, complex shapes from simple polygon meshes, widely used in computer graphics and animation.
ARTICLE PAGE
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