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Spline curves: Piecewise polynomial curves which are maximally smooth:
Polynomial degree continuous

Constructing Spline curves

• Degree 1: Linear segments.
Connect endpoints for continuity.
• Degree 2: Quadratic segments.
Connect endpoints for continuity.Use identical vector for first and last edge of subsequent segments for continuity.
• Degree 3: Cubic segments.
Connect endpoints for continuity.Use identical vector for first and last edge of subsequent segments for continuity.Extend the middle edges of both segments and find the intersection point: If the so-created A-frame splits each pair of collinear edges in a 1:1 ratio, the connection between the segments is continuous.

This allows to create a cubic spline curve by only specifying A-frame vertices:

• Connect A-frame vertices by edges and split 1:1:1
• Add edges between each first and last vertex between neighboring edges and split 1:1
• Connect each 4 vertices with one cubic spline