That's right the Hobby splines are a strict subset of the polynomial cubic Béziers.
The polynomial cubic Béziers are defined by 4 control points. In 2D that's 4 x 2 = 8 degrees of freedom. In the terminology of HN raphlinus's PhD thesis, we fix the 2 endpoints, which removes 4 degrees of freedom, and they're referred to as a 4-parameter family.
In addition to the 2 endpoints, Hobby splines are defined by the start and end tangent angles Θ0 and Θ1, plus a tension parameter τ.
If we fix the 2 endpoints and the tension parameter τ, and allow Θ0 and Θ1 to vary, then we have a 2-parameter subspace of the 4-parameter space of polynomial cubic Béziers for 2 fixed endpoints.
If we allow the tension parameter τ to vary we have a 3-parameter subspace.
So we have lost 1 degree of freedom, but in terms of aesthetics we haven't lost so much because the majority of the splines that we've thrown away are likely uglier than the splines we retained.
One TL/DR from HN raphlinus's thesis: Euler spiral splines are about the best you can do for a 2-parameter family that cover all reasonable possibilities for Θ0 and Θ1 with good aesthetics.
Comments
That's right the Hobby splines are a strict subset of the polynomial cubic Béziers.
The polynomial cubic Béziers are defined by 4 control points. In 2D that's 4 x 2 = 8 degrees of freedom. In the terminology of HN raphlinus's PhD thesis, we fix the 2 endpoints, which removes 4 degrees of freedom, and they're referred to as a 4-parameter family.
In addition to the 2 endpoints, Hobby splines are defined by the start and end tangent angles Θ0 and Θ1, plus a tension parameter τ.
If we fix the 2 endpoints and the tension parameter τ, and allow Θ0 and Θ1 to vary, then we have a 2-parameter subspace of the 4-parameter space of polynomial cubic Béziers for 2 fixed endpoints.
If we allow the tension parameter τ to vary we have a 3-parameter subspace.
So we have lost 1 degree of freedom, but in terms of aesthetics we haven't lost so much because the majority of the splines that we've thrown away are likely uglier than the splines we retained.
One TL/DR from HN raphlinus's thesis: Euler spiral splines are about the best you can do for a 2-parameter family that cover all reasonable possibilities for Θ0 and Θ1 with good aesthetics.