Fitting a curve within a triangle

[jenny_wui]

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?

[skeezix6]

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

[jenny_wui]

Thank you very much for the suggestion.