The text describes a deep dive into the mathematics behind animations, specifically the use of Fourier series to create complex animations. The speaker explains that each vector in the animation rotates at a constant integer frequency. When these vectors are combined, they create a shape over time. The initial size and angle of each vector can be adjusted to create any shape you want.
The animation in question features 300 rotating arrows. The speaker points out the simplicity of each individual arrow's action, which is rotation at a steady rate. However, when combined, they create a complex and mind-boggling pattern. The speaker notes that the complexity in the evolution of this pattern is captured in the difference in the decay rates for the different pure frequency components.
The speaker then moves on to explain Fourier series, which are often described as functions of real numbers broken down as a sum of sine waves. The speaker explains that Fourier series are more general than this, and can be applied to functions whose output can be any complex number in the 2D plane.
The speaker then explains that the key challenge is to find the coefficients that make up the Fourier series. The speaker notes that for the computations, they will be thinking about functions whose input is some real number on a finite interval, and the output can be any complex number in the 2D plane.
The speaker then explains that the Fourier series is essentially a sum of little vectors all rotating at some constant integer frequency. The speaker notes that the complexity in these decompositions is entirely captured in a little expression.
In conclusion, the Fourier series is a powerful tool for breaking down complex animations into simpler components. This allows for precise control over the initial size and direction of each vector, which in turn creates the shape of the animation.
1. The text discusses the math behind an animation, specifically a complex Fourier series, where each vector rotates at a constant integer frequency. When these vectors are added together, they draw out a shape over time by adjusting the initial size and angle of each vector.
2. The animation has 300 rotating arrows in total. The intricacy of the animation is due to the mind-boggling complexity of the collection of arrows, which trace out a specific shape.
3. The Fourier series, a mathematical concept, is used to describe the evolution of the temperature distribution on a rod over time. The series involves adding up exponentially decaying cosine waves to construct a solution for a new tailor-made initial condition.
4. The Fourier series can be used to express any initial distribution as a sum of sine waves, even complex ones. This is achieved by considering infinite sums of these waves, which can approximate a discontinuous flat function.
5. The Fourier series is used to solve a partial differential equation (PDE) with a discontinuous initial condition. The solution involves finding the coefficients of the infinite sum, which describe how the step function will evolve over time.
6. The Fourier series can be applied to functions whose output can be any complex number in the 2D plane. This makes the computations cleaner and easier to understand.
7. The Fourier series involves the complex exponential e to the I times T, which walks around the unit circle at a rate of one unit per second. This is used to describe rotating vectors in the Fourier series.
8. The Fourier series allows for the control of the initial size and direction of each number in the series by multiplying it by some complex constant. This is done by finding the average of the function over the whole series, which kills all terms other than the one being considered.