$$a_n = \sqrt[4]{ 2 } \cdot \sqrt[4]{ s \cdot ( i - s ) - 2 \cdot r^2 } \quad \text{...Numerator}.$$
$$a_d = \sqrt{ s \cdot ( 5 \cdot s - 3 \cdot i ) + 4 \cdot r^2 } \quad \text{...Denominator.}$$
$$a = \dfrac{a_n}{a_d}$$
$$y = a^2 \cdot ( x - s )^2$$
Circular
Input values:
$$w \quad \text{...Gusset width}$$
$$g \quad \text{...Knuckle radius}$$
Compute gusset radius:
The radius of the circle, that is tangential to the knuckle cylinder.
$$r = \dfrac{ 2 \cdot g \cdot w \space + \space w^2}{ 2 \cdot g}$$
Compute gusset height:
The point of intersection between the knuckle cylinder and the gusset cutter.
$$h = \dfrac{ g \cdot r }{ \sqrt{ 2 \cdot g \cdot w \space + \space g^2 \space + \space r^2 \space + \space w^2 } }$$
Compute the intersection point between the knuckle and gusset cutting tool, using gusset height:
The coordinates of the intersection point are, $p(x,h)$, where $h$ is the vertical value of the coordinate.
$$x = \dfrac{ h \cdot ( g \space + \space w ) }{ r }$$
Linear
Input values.
$$w \quad \text{...Gusset width.}$$
$$h \quad \text{...Hinge leaf height.}$$
$$r \quad \text{...Knuckle radius is equal to the leaf gauge.}$$
$$r \quad \text{...Leafe gauge.}$$
Compute gusset tangent gradient.
$$s = w + r \quad \text{...Cartesian position of the point where the gusset curve merges with the leaf.}$$
