The Cubic Ease In Function

Oct 13, 2025 by talynt.org 4 views

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Introduction

Easing functions are used in animation and game development to create smooth and natural-looking movements. They are often used to control the speed of an object's movement over time, making it more visually appealing. However, when working with easing functions, it's not uncommon to encounter situations where we need to find the derivative of the function. The derivative of an easing function represents the rate of change of the function with respect to time, which is essential in many applications, such as physics engines and motion planning algorithms.

In this article, we will explore how to find the derivative of an easing function that accounts for duration and distance. We will use C# as our programming language and provide a step-by-step guide on how to derive the derivative of a given easing function.

What is an Easing Function?

An easing function is a mathematical function that describes how an object's movement changes over time. It's a way to control the speed and acceleration of an object's movement, making it more natural and visually appealing. Easing functions are often used in animation, game development, and other fields where motion is a critical aspect.

There are many types of easing functions, including linear, quadratic, cubic, and quartic, among others. Each type of easing function has its own characteristics and is used in different situations.

The Cubic Ease In Function

Let's take a look at the cubic ease in function, which is a common easing function used in animation and game development.

public static float CubicEaseIn(float currentTime, float startValue, float endValue, float duration)
{
    return endValue * (currentTime /= duration) * (currentTime - 1) * (currentTime - 1);
}

This function takes four parameters: currentTime, startValue, endValue, and duration. The currentTime parameter represents the current time, startValue and endValue represent the starting and ending values of the movement, and duration represents the total duration of the movement.

Finding the Derivative of the Cubic Ease In Function

To find the derivative of the cubic ease in function, we need to apply the power rule of differentiation, which states that if f(x) = x^n, then f'(x) = nx^(n-1).

Let's start by rewriting the cubic ease in function in a more general form:

f(t) = endValue * (t / duration) * (t - 1) * (t - 1)

Now, let's apply the power rule of differentiation to find the derivative of the function:

f'(t) = endValue * (1 / duration) * (t / duration) * (t - 1) * (t - 1) + endValue * (t / duration) * (2 * (t - 1))

Simplifying the expression, we get:

f'(t) = endValue * (1 / duration) * (t / duration) * (t - 1) * (t - 1) + endValue * (2 * t / duration) * (t - 1)

Implementing the Derivative of the Cubic Ease In Function in C#

Now that we have found the derivative of the cubic ease in function, let's implement it in C#:

public static float CubicEaseInDerivative(float currentTime, float startValue, float endValue, float duration)
{
    return endValue * (1 / duration) * (currentTime / duration) * (currentTime - 1) * (currentTime - 1) + endValue * (2 * currentTime / duration) * (currentTime - 1);
}

This function takes the same parameters as the cubic ease in function and returns the derivative of the function at the given time.

Conclusion

In this article, we have explored how to find the derivative of an easing function that accounts for duration and distance. We have used the cubic ease in function as an example and applied the power rule of differentiation to find the derivative of the function. We have also implemented the derivative of the cubic ease in function in C#.

The derivative of an easing function is an essential concept in many applications, such as physics engines and motion planning algorithms. By understanding how to find the derivative of an easing function, developers can create more realistic and natural-looking movements in their applications.

Future Work

In the future, we can explore other types of easing functions and find their derivatives. We can also apply the derivative of an easing function to real-world problems, such as motion planning and physics engines.

References

[1] Wikipedia: Easing function
[2] Wikipedia: Derivative
[3] C# documentation: Math class

Code

using System;

public class EasingFunctions
{
    public static float CubicEaseIn(float currentTime, float startValue, float endValue, float duration)
    {
        return endValue * (currentTime /= duration) * (currentTime - 1) * (currentTime - 1);
    }

    public static float CubicEaseInDerivative(float currentTime, float startValue, float endValue, float duration)
    {
        return endValue * (1 / duration) * (currentTime / duration) * (currentTime - 1) * (currentTime - 1) + endValue * (2 * currentTime / duration) * (currentTime - 1);
    }
}

Note: The code provided is a simple implementation of the cubic ease in function and its derivative. In a real-world application, you would need to consider factors such as edge cases and performance optimization.

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Introduction

In our previous article, we explored how to find the derivative of an easing function that accounts for duration and distance. We used the cubic ease in function as an example and applied the power rule of differentiation to find the derivative of the function. In this article, we will answer some frequently asked questions about finding the derivative of an easing function.

Q: What is the purpose of finding the derivative of an easing function?

A: The derivative of an easing function represents the rate of change of the function with respect to time. It's essential in many applications, such as physics engines and motion planning algorithms, where you need to calculate the acceleration or velocity of an object.

Q: How do I find the derivative of an easing function?

A: To find the derivative of an easing function, you need to apply the power rule of differentiation. This rule states that if f(x) = x^n, then f'(x) = nx^(n-1). You can also use the chain rule and the product rule of differentiation to find the derivative of a more complex easing function.

Q: What are some common easing functions that I can find the derivative of?

A: Some common easing functions that you can find the derivative of include:

Linear easing function: f(t) = t
Quadratic easing function: f(t) = t^2
Cubic easing function: f(t) = t^3
Quartic easing function: f(t) = t^4

Q: How do I implement the derivative of an easing function in code?

A: To implement the derivative of an easing function in code, you can use a programming language such as C# or Java. You can use the power rule of differentiation to find the derivative of the easing function and then implement it in code.

Q: What are some real-world applications of finding the derivative of an easing function?

A: Some real-world applications of finding the derivative of an easing function include:

Motion planning: Finding the derivative of an easing function can help you calculate the acceleration or velocity of an object in motion planning algorithms.
Physics engines: Finding the derivative of an easing function can help you simulate the motion of objects in physics engines.
Animation: Finding the derivative of an easing function can help you create smooth and natural-looking animations.

Q: What are some edge cases that I should consider when finding the derivative of an easing function?

A: Some edge cases that you should consider when finding the derivative of an easing function include:

Division by zero: When finding the derivative of an easing function, you may encounter division by zero. You should handle this case carefully to avoid errors.
Negative values: When finding the derivative of an easing function, you may encounter negative values. You should handle this case carefully to avoid errors.
Infinity: When finding the derivative of an easing function, you may encounter infinity. You should handle this case carefully to avoid errors.

Q: How do I optimize the derivative of an easing function for performance?

A: To optimize the derivative of an easing function for performance, you can use techniques such as:

Caching: You can cache the derivative of the easing function to avoid recalculating it every time.
Memoization: You can use memoization to store the derivative of the easing function and reuse it when needed.
Parallel processing: You can use parallel processing to calculate the derivative of the easing function in parallel.

Conclusion

In this article, we have answered some frequently asked questions about finding the derivative of an easing function. We have discussed the purpose of finding the derivative of an easing function, how to find the derivative of an easing function, and some common easing functions that you can find the derivative of. We have also discussed some real-world applications of finding the derivative of an easing function and some edge cases that you should consider when finding the derivative of an easing function. Finally, we have discussed how to optimize the derivative of an easing function for performance.

Future Work

References

[1] Wikipedia: Easing function
[2] Wikipedia: Derivative
[3] C# documentation: Math class

Code

using System;

public class EasingFunctions
{
    public static float CubicEaseIn(float currentTime, float startValue, float endValue, float duration)
    {
        return endValue * (currentTime /= duration) * (currentTime - 1) * (currentTime - 1);
    }

    public static float CubicEaseInDerivative(float currentTime, float startValue, float endValue, float duration)
    {
        return endValue * (1 / duration) * (currentTime / duration) * (currentTime - 1) * (currentTime - 1) + endValue * (2 * currentTime / duration) * (currentTime - 1);
    }
}