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Was looking at this video https://youtu.be/60VoL-F-jIQ?t=213 and a linear interpolation between two equations is mentioned. I don't understand how one linearly interpolates two equations. The video shows the following 2 equations and a 3rd one which is said to be the linear interpolation of the first two.

  1. $$a(x) = x^2$$
  2. $$b(x) = -(x-1)^2 + 1$$
  3. $$ c(x) = a(x)\cdot(1-x) + b(x) \cdot x$$

I don't understand how equation 3 was reached from equations 1 and 2?

user170231
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user21279755
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    The idea is that $c(x)$ is a function that behaves more or less like one function at one endpoint, and more or less like the other function at the other endpoint. Notice how $c(0)=a(0)$ and $c(1)=b(1)$. For values of $x$ close to $0$, the interpolating function $c(x)$ will more closely resemble $a(x)$; for values of $x$ close to $1$, $c(x)$ will look more like $b(x)$. – user170231 Apr 10 '23 at 16:46
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    It's simply a definition: there is nothing to understand. The interpolation of two functions $a(x), b(x)$ is defined as $$a(x) \cdot (1-x) + b(x) \cdot x$$ – Crostul Apr 10 '23 at 18:25
  • Seems similar to $a(x) (1-\lambda) +b(x)\lambda$ for $\lambda \in (0,1)$. Though it's just a definition as pointed @Crostul – Qise Apr 10 '23 at 19:07
  • I was starting feel like I was not good enough for not being able to derive the equation, but if it's just a definition, then ok. Thank you :) – user21279755 Apr 10 '23 at 23:01

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Let's assume a(x),b(x) corresponds to N1,N2 respectively. Also if in the interpolation we use ξ instead of x below you can see the interpolation between the 2 functions which is nothing else than interpolation between 2 numbers actually along ξ[0,1]. enter image description here

Manos
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