Combining tessellation families: example

Variable crease pattern (slider controls):

H. A. Verrill

In my 2021 Bridges paper, and also at the 2020 cfc origami creators event, I talked about families of deformations of the waterbomb base tessellation, and also about how or whether it might be possible to combine two one parameter families of deformations to create one two parameter family.
This page gives an example of another situation of combining two parameters, which I didn't cover in my talk. The first family on this page is the two peak waterbomb variation, which is on my multipeak waterbomb base tessellation page, and is obtained with b=c=0. The other family is the dragon scales variation, which is one of the families on my page combining LamiOri and dragon scales tessellations.
If you want a challenge, before looking at the program below, look at the following two pictures, and try and work out how to combine them into one family with two parameters. That is, given a family with a as a parameter, and another family with b as a parameter, as shown in these diagrams, is there a family with two parameters, which restricts to these two families when one parameter is zero? Or, if not zero, if we restrict to some one dimensional subspace of the 2-dimensional parameter space. To make this question non trivial, we require that there are sides of the parameterised unit which have lengths a and b. A possible solution is given below. In this solution, the lengths do not change direction, though I do not know if this should be a requirement for a solution, or if other solutions without this condition exist, or other solutions with this condition. My question is still not completely precisely defined.... I have an answer looking for the right question.

Parameters of two one parameter families:
Two peak unit parameter
a: 0
Dragon scale parameter
b: 0
2D Parameter of combined family:
a1
a2
0 ≤ a1 + a2 ≤ 1
alternative view: a replaced by a1 and a2
a1: 0
a2: 0
Other (non crease pattern) parameters for image
scale: 1
shift x-coord: 0
shift y-coord: 0
line width: 1
Colour variables:
var 1: 0
var 2: 0
var 3: 0.5
var 4: 0