Biarcs are commonly used in geometric modeling and computer graphics. They can be used to approximatesplines and other plane curves by placing the two outer endpoints of the biarc along the curve to be approximated, with a tangent that matches the curve, and then choosing a middle point that best fits the curve. This choice of three points and two tangents determines a unique pair of circular arcs, and the locus of middle points for which these two arcs form a biarc is itself a circular arc. In particular, to approximate a Bézier curve in this way, the middle point of the biarc should be chosen as the incenter of the triangle formed by the two endpoints of the Bézier curve and the point where their two tangents meet. More generally, one can approximate a curve by a smooth sequence of biarcs; using more biarcs in the sequence will in general improve the approximation's closeness to the original curve.
In the below examples biarcs are subtended by the chord and is the join point. Tangent vector at the start point is , and is the tangent at the end point
Fig. 2 shows six examples of biarcs
Biarc 1 is drawn with Biarcs 2-6 have
In examples 1, 2, 6 curvature changes sign, and the join point is also the inflection point. Biarc 3 includes the straight line segment .
Biarcs 1–4 are short in the sense that they do not turn near endpoints. Alternatively, biarcs 5,6 are long: turning near one of endpoints means that they intersect the left or the right complement of the chord to the infinite straight line.
Biarcs 2–6 share end tangents. They can be found in the lower fragment of Fig. 3, among the family of biarcs with common tangents.
Fig. 3 shows two examples of biarc families, sharing end points and end tangents.
Fig. 4 shows two examples of biarc families, sharing end points and end tangents, end tangents being parallel:
Fig. 5 shows specific families with either or
Fig. 2. Examples of biarcs
Fig. 3. Biarcs families with common tangents (two examples)
Fig. 4. Biarcs families with parallel end tangents
Fig 5. Biarcs families with either or
Different colours in figures 3, 4, 5 are explained below as subfamilies
In particular, for biarcs, shown in brown on shaded background (lens-like or lune-like), the following holds:
the total rotation (turning angle) of the curve is exactly (not , which is the rotation for other biarcs);
: the sum is the angular width of the lens/lune, covering the biarc, whose sign corresponds to either increasing (+1) or decreasing curvature (-1) of the biarc, according to generalized Vogt's theorem (ru).