Any equation that fits in this shape? (variable concavity)

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In summary, the conversation discusses the need for an equation that can create lines with varying levels of concavity and convexity, passing through the points (0,1) and (1,0) and having a single coefficient parameter. An equation for a circle with a center of (b,b) and a radius that crosses through the two points is suggested, but it is determined that only the bottom half of the circle is needed. Another equation, (1-x)^a + (1-y)^a = 1, is proposed as a solution, with a=1 resulting in a straight line and increasing values of a creating curves closer to the origin. This equation also exhibits symmetry.
  • #1
I'm working on an audio synthesis project and I need an equation that can create these lines:


That is, I would like an equation that describes a line that always passes through (0,1) and (1,0), and has a single coefficient parameter that can be varied to create functions of intermediate concavity/convexity like the red line shown.

The concavity/convexity should be symmetric as if mirrored along the line of y=x between the boundaries of (0,1) and (1,0). Anything outside that range is irrelevant.

Is there an equation that can do this?


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  • #2
If I set the center of circle to be (x,y) = (b,b) for b>=1, and then set the radius of the circle to cross through (1,0) and (0,1):
r = sqrt(bb+ (bb-1)) = sqrt(2bb - 1)
then the equation for the circle will be:
(x-b)^2 + (y-b)^2 = bb +(b-1)^2
xx-2xb+bb+yy-2yb+bb = 2bb-2b+1
xx-2xb+bb+yy-2yb+bb-2bb+2b-1 = 0
xx-2xb+yy-2yb+2b-1 = 0
y = (2b+/-sqrt(2bb-4(xx-2xb+2b-1)))/2
y = b+/-sqrt(bb-(xx-2xb+2b-1))

but we only want the bottom of the circle, so:
y = b-sqrt(bb-(xx-2xb+2b-1))
  • #3
##(1-x)^a + (1-y)^a = 1## - this can be solved for y if you prefer that: ##y=1-(1-(1-x)^a)^{1/a}##

a=1 gives a straight line, a=2 gives a quarter-circle, larger a give something that passes closer to (0,0). It is always symmetric as you can see from the first equation.
Example curves.

1. What is the significance of variable concavity in equations?

Variable concavity refers to the changing curvature of a function or equation. It can provide important information about the behavior of the function, such as the existence of maximum or minimum points.

2. How is variable concavity determined in an equation?

Variable concavity can be determined by taking the second derivative of the function and analyzing its sign changes. A positive second derivative indicates a concave up shape, while a negative second derivative indicates a concave down shape.

3. Can an equation have both concave up and concave down sections?

Yes, an equation can have both concave up and concave down sections, as long as its second derivative changes sign at some point. This is known as a point of inflection.

4. How does variable concavity affect the graph of an equation?

The variable concavity of an equation can significantly impact its graph. For example, a function with a concave up section will have a U-shaped graph, while a function with a concave down section will have an inverted U-shaped graph. These changes in curvature can also affect the location of any maximum or minimum points.

5. Can variable concavity be used to optimize an equation?

Yes, variable concavity can be used to optimize an equation by identifying the points of maximum or minimum value. By analyzing the concavity of the equation, we can determine whether a point is a maximum or minimum and use that information to optimize the function.

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