Joshua Deakin

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2016.03.14 - Shape Functions - Linear Transition

Using linear interpolation to transition between shape functions.

Linear interpolation function between a and b.

The transition is made by changing the value of t within the range:
{ 0 <= t <= 1 }

result = (1 - t) * a + t * b
where:
t=0 => result = a
t=1 => result = b

This can be used to transition between two shape functions, a and b:

a = f_1(d_1, d_2, ..., d_n)

b = f_2(d_1, d_2, ..., d_n)

result = (1 - t) * f_1(d_1, d_2, ..., d_n) + t * f_2(d_1, d_2, ..., d_n)

Where:
{ 0 <= t <= 1 }

d_1 = x
d_2 = y
d_n = highest_dimension

Finally, this can be generalised to allow blends between N shape functions:

T = t_1 + t_2  + t_3 + ... + t_N

T = 1 at any transition position.

Where N is the number of shape functions.

result = t_1 * f_1 + t_2 * f_2 + t_3 * f_3 + ... + t_N * f_N
(or)
result =
t_1 * f_1(d_1, d_2, ..., d_n) +
t_2 * f_2(d_1, d_2, ..., d_n) +
t_3 * f_3(d_1, d_2, ..., d_n) +
... + t_N * f_N(d_1, d_2, ..., d_n)
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