# Sum w.r.t Y axis of two Bézier curves with different control points

Suppose two 2D Bézier curves of the same degree but different control points. $\gamma_1(t) = \Sigma_{i=0}^n B_{i,n}(t) q_i$ and $\gamma_2(t) = \Sigma_{i=0}^n B_{i,n}(t) Q_i$

where $q_0 = Q_0 = (0;0), q_n = Q_n = (1;0)$ and $q_i \neq Q_i$

Let $y_1(x)$ and $y_2(x)$ be their Cartesian forms.

Suppose a third curve which is the sum of the two previous curves w.r.t Y axis i.e. $y_3(x) = y_1(x) + y_2(x)$.

Is it possible to express $y_3(x)$ as a parametric curve $\gamma_3(t)$ using either Bézier curves, B-splines or even NURBS ?

• I don't understand something: it may be that the curves can't be parameterized by $x$ – mathcounterexamples.net Jan 8 '18 at 20:24
• Bezier curves are defined by their control points. So the new parametric curve is just defined by the new control points i.e., the ones calculated by the sum of the desired control points of the original two curves. Am I missing something? How are these "cartesian forms" defined? – Mauricio Cele Lopez Belon Jan 8 '18 at 21:14
• Perhaps what you want is to fit a bezier curve to some cartesian points obtained in some way from two bezier curves? – Mauricio Cele Lopez Belon Jan 8 '18 at 21:31
• @mathcounterexamples.net Actually they are , the first curve is parametrized as $x_1 (t) = t^2$ and $x_2(t) = (1-t)^2$ – FineUser Jan 8 '18 at 22:35
• @Mauricio Cele Lopez Belon. No, the resulting curve is defined as the sum of each two points of the other curves having the same x (not the same t). – FineUser Jan 8 '18 at 22:39