Question

I don't see how:
if \(r_1(t) = (1-t)P+t (1,0) ~~~0\leq t\leq1\)
\(r_2(t) =(1-t)(1,0) + tQ~~~0\leq t\leq 1\)
where P, Q are initial and terminal points respectively,
then \((r_1+r_2)(t)=\begin {cases}r_1(2t) ~~~0\leq t\leq 1/2\\r_2(2t-1)~~1/2\leq t\leq 1\end {cases}\)
Please, help

Answer 1

@dan815

Answer 2

@freckles

Answer 3

if we have r(t)=f(t) on the interval 0<t<1
then r(2x)=f(2x) on the interval 0<2x<1
so that would mean 0<x<1/2

or if we had r(2x-1)=f(2x-1) the 0<2x-1<1 so 1<2x<2 then 1/2<x<1

Answer 4

is that what you are asking about the intervals?

Answer 5

or the whole thing

Answer 6

the whole. :)

Answer 7

Well I understand how they got the intervals
but the other parts...

Answer 8

\[(r_1+r_2)(t)=(1-t)P+t(1,0)+(1-t)(1,0)+tQ\]

Answer 9

=P -tP +(1,0) +tQ :)

Answer 10

\[r_1(2t)=(1-2t)P+2t(1,0)\]

Answer 11

but the interval is changed.

Answer 12

\[r_2(2t-1)=(1-(2t-1))(1,0)+(2t-1)Q \\ = (-2t+2)(1,0)+(2t-1)Q\]

Answer 13

and yeah so the interval is changed ...

Answer 14

stupid question
(1,0) means what exactly

Answer 15

hmmm interesting

Answer 16

oh wait

Answer 17

(1,0) is a point and Q is a point

Answer 18

so r_2 is a point right?

Answer 19

r_1 is a point

Answer 20

Those are from my Prof's lecture. I don't get this part. :)

Answer 21

it looks like they just translated r_1 and r_2

Answer 22

\[r_1(2t)=(1-2t)P+2t(1,0) , 0 \le t \le \frac{1}{2} \equiv r_1(t)=(1-t)P+t(1,0),0\le t \le 1\]
(1-t) is the same as (1-2x) when t=0 and x=0
(1-t) is the same (1-2x) when t=1 and x=1/2

Answer 23

this is original problem:

Answer 24

I plugged in 1/4
one way I got
\[(1,0)+\frac{3}{4}P+\frac{1}{4}Q \\ \]
and the other way I got
\[\frac{1}{2}P+\frac{1}{2}(1,0)\]
If i didn't make a sense. I don't know how to tell if these are the same :(

Answer 25

if i didn't make a mistake*

Answer 26

\[\frac{1}{4}Q=-\frac{1}{2}(1,0)-\frac{1}{4}P \\ Q=-2(1,0)-P?\]

Answer 27

aren't you just changing the parameter so that t = (0,1) gives you two sequential paths :

0->1/2 : r1
1/2 -> 1: r2

Answer 28

Please, explain more.@ganeshie8

Answer 29

if you think of "t" as time and both the paths as the walk of bugs :

Answer 30

the first parameterization gives you two parallel paths : two bugs starting at different positions and travelling as described by the path