I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and

George Bray

George Bray

Answered question

2022-06-13

I'm currently reading Pugh's Analysis. He makes the statement that the line between two points x and y is the set of linear combinations s x + t y where s + t = 1. I'm satisfied that this is true, as the line between two points x and y has the equation y x 2 = x 2 y 2 x 1 y 1 ( x x 1 ) and the points of form ( s x 1 + t y 1 , s x 2 + t y 2 ) are solutions of the equation. I still don't have any reasonable geometric intuition for why this is true though.

Answer & Explanation

Ryan Newman

Ryan Newman

Beginner2022-06-14Added 26 answers

It’s easiest to see what’s going on when 0 s , t 1. In that case s x + t y is a weighted mean of x and y. When s = t = 1 2 , for instance, it’s the ordinary arithmetic mean, and the point 1 2 x + 1 2 y is the midpoint of x y ¯ . When s = 1 3 and t = 2 3 , y is given twice the weight of x, so the point is ‘twice as close’ to y as it is to x: in more understandable terms, it’s half as far from y as it is from x, so that it’s 2 3 of the way from x to y. In general, s x + t y is t fraction of the way from x to y when 0 s , t 1.
Once s and t get outside the range [ 0 , 1 ], it’s harder to get an intuitive picture, but you can still regard s x + y t is a kind of weighted mean in which one of the weights is negative. Thus, x + 2 y = y + ( y x ) is the point that is y x units past y in the direction away from x, and in general s x + t y is the point that is y x units past ( t 1 ) y in the direction away from x when t > 1. (And of course everything just turns around when s > 1.)

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