# Linearity of Power of Point

1)What is meant by $$\mathbb{R}^2 \rightarrow \mathbb{R}$$ in this context?

2)What is meant by $$C=kA+(1−k)B$$?Aren't $$A,B,C$$ just points in euclidean plane?How can they be related by an equation?

3)Why is it sufficient to show the condition for points $$A$$ and $$B$$?

4)What is meant by $$F$$ is linear?How to prove it and what is it's use?

Sorry if that's too obvious.I am not aware of the terminology used here i guess.

1)What is meant by R2→R in this context?

$$F$$ is a function whose domain is $$\mathbb{R}^2$$ and codomain is $$\mathbb{R}$$. The domain is $$\mathbb{R}^2$$ because its inputs are points, and the codomain is $$\mathbb{R}$$ because its outputs are real numbers. Since what $$F$$ does is ignore the $$\omega$$ terms in $$\mathbb{P}$$, it is what is called a `partial function' in programming. Put another way, the domain of $$\mathbb{P}$$ is $$\mathbb{R}^2 \times \mbox{the space of circles}^2$$ (sorry, I don't know what space $$\omega$$ lives in), and its codomain is $$\mathbb{R}$$. What $$F$$ does is ignore some of the inputs.

2)What is meant by C=kA+(1−k)B?Aren't A,B,C just points in euclidean plane?How can they be related by an equation?

It's a statement that $$C$$ is on the line segment between $$A$$ and $$B$$. This kind of formula: $$kx + (1-k)y$$ is called a convex combination of $$x$$ and $$y$$. In the context of two points in Euclidean space, it is a way to associate the points on the line thru $$x$$ and $$y$$ with values of $$k$$. If $$0 \leq k\leq 1$$, then the point is between $$x$$ and $$y$$.

3)Why is it sufficient to show the condition for points A and B?

4)What is meant by F is linear?How to prove it and what is it's use?

The answers to 3 and 4 are connected. A function $$f$$ is linear if $$f(a + cb) = f(a) + c f(b)$$ no matter what $$a$$ and $$b$$ (and $$c$$) are. So the statement about sufficiency is just a bit of semantic fluff to get the proof started. Any time you want to prove a function is linear, you start with a statement similar to: "Let $$a$$ and $$b$$ be in the domain of the function".

• Thanks for the help but in the 1st point i know the domains and codomains are $\mathbb{R}^2$ and $\mathbb{R}$.But what does it mean for $\mathbb{R}^2$ and $\mathbb{R}$ here?I mean what role does $P$ play as $\mathbb{R}^2$ and $\mathbb{P}(P, \omega)$ as $\mathbb{R}$? Mar 19, 2021 at 13:48
• OK, I think I understand your question. I'll edit my answer. Mar 19, 2021 at 13:54
• I don't think I'd say $F$ "ignores" the circles. Rather, to get $F$ we first have to fix the choice of the two circles so that the only parameter of $\mathbb P$ that is allowed to vary is $P$. Moreover, I would call $F$ a partially applied function. A partial function is one that does not produce output values for every value in its domain. Mar 19, 2021 at 14:17
• Thanks again but i have one last inquiry.$f$ is linear if $f(a+cb)=f(a)+cf(b)$.What are $a,b,c$ according to the context?I mean the proof mentioned in the post? Mar 19, 2021 at 14:30
• @DavidK in programming this is indeed what's called a partial function. See for example: docs.python.org/3/library/functools.html. As for the ignores / fixes distinction, yes I agree that's also a fine way to think about it. Mar 19, 2021 at 19:32