| Safe Haskell | Safe |
|---|---|
| Language | Haskell2010 |
Imj.Graphics.Math.Ease
Contents
- invQuartEaseInOut :: Float -> Float
- discreteAdaptor :: (Float -> Float) -> Int -> Float -> Float
- discreteInvQuartEaseInOut :: Int -> Float -> Float
4th order inverse easing, continuous
Easing is traditionally seen as a function from time to value.
Here, it is a function from value to time, hence the use of the term Inverse in the title.
Returns the time \( t \in [\,0,1]\, \) at which a value \( y \in [\,0,1]\,\) is reached given a 4th order ease in-out function \( quartEaseInOut \):
\[ y = quartEaseInOut(t) = \begin{cases} {1 \over 2} * (2*t)^4, & \;\;\;\; \text{if $t < {1 \over 2}$} \\[2ex] -{1 \over 2} * \left( [ 2*(t-1) ]^4 - 2 \right), & \;\;\;\; \text{if $t > {1 \over 2}$} \end{cases} \]
To find the formulas of invQuartEaseInOut, we need to invert \( quartEaseInOut \),
i.e. we need to express \(t\) in terms of \(y\):
\[ \text{$quartEaseInOut$ is strictly increasing} \implies \begin{cases} t<{1 \over 2} \iff y<{1 \over 2} \\ t>{1 \over 2} \iff y>{1 \over 2} \end{cases} \]
\[ \begin{alignedat}{3} \text{if $y < {1 \over 2} $, given the $quartEaseInOut$ equation for $t < {1 \over 2} $ :} && y &= {1 \over 2} * (2*t)^4 && \\ \implies && \quad t &= \left({y \over 2^3}\right)^{1/4} && \quad \forall y < {1 \over 2} \\ \text{if $y > {1 \over 2} $, given the $quartEaseInOut$ equation for $t > {1 \over 2} $ :} && y &= - {1 \over 2} * \left( [2*(t-1)]^4 - 2 \right) && \\ \implies && \quad t &= 1-\left[{1-y \over 2^3}\right]^{1/4} && \quad \forall y > {1 \over 2} \end{alignedat} \]
Note that there are multiple solutions, we chose the ones that produce results in the \( [\,0,1]\, \) range.
Hence, the formulas for invQuartEaseInOut are :
\[ t = invQuartEaseInOut(y) = \begin{cases} \left({y \over 2^3}\right)^{1/4}, & \text{if $y < {1 \over 2}$} \\[2ex] 1-\left[{1-y \over 2^3}\right]^{1/4}, & \text{if $y > {1 \over 2}$} \end{cases} \]
From continuous to discrete
Easing in a continuous world is easy (no pun intended), but easing in a discrete world is harder : we have to make sure the discretization will not break the visual easing effect.
The discreteAdaptor function does just that, making a continuous easing
function usable in a discrete context.
Arguments
| :: (Float -> Float) | Continuous (optionally inverse) ease in/out function |
| -> Int | The number of discrete steps |
| -> Float | Input value |
| -> Float | (optionnaly inverse) Eased value |
Adapts continuous inout ease functions to the discrete case.
4th order inverse easing, discrete
Using discreteAdaptor on invQuartEaseInOut we can make
discreteInvQuartEaseInOut :