Fitting a curve within a triangle


I have drawn an ellipsoid which passes through base BC of the triangle. Some part of the ellipsoid is outside the triangle. I want the curve remain inside the triangle. I know ellipsoid is not a solution. How can I modify the curve so that it will pass through the base but will remain inside the triangle ABC? I am not sure whether parabola may help to make front tapered in shape. But how to make it pass through the base? Or any other curve fitting suggestion available?

Cubic Hermite spline. Set the end points to the vertices of the base, and the slopes parallel with the triangle edges.

Thank you very much for the suggestion.