How to map interval [0, 100] to the interval [100, 350]?

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I have an interval $[0; 100]$ and would like to map it to this new interval: $[100;350]$.



I thought about multiplying it by $3.5$, but that would give the interval $[0;350]$. And adding to each of these elements $100$ would give: $[100;450]$. Hence my question: is it possible to do what I want?



Note that I can settle for the interval $[0;350]$ : in my program, it will be enough if I exclude the numbers present in the interval $[0;99]$.






interval-arithmetic







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  • 7




    how about multiplying by 2.5 instead?
    – Lord Shark the Unknown
    19 hours ago














up vote
3
down vote

favorite
1












I have an interval $[0; 100]$ and would like to map it to this new interval: $[100;350]$.



I thought about multiplying it by $3.5$, but that would give the interval $[0;350]$. And adding to each of these elements $100$ would give: $[100;450]$. Hence my question: is it possible to do what I want?



Note that I can settle for the interval $[0;350]$ : in my program, it will be enough if I exclude the numbers present in the interval $[0;99]$.






interval-arithmetic







share|cite|improve this question



















  • 7




    how about multiplying by 2.5 instead?
    – Lord Shark the Unknown
    19 hours ago












up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





I have an interval $[0; 100]$ and would like to map it to this new interval: $[100;350]$.



I thought about multiplying it by $3.5$, but that would give the interval $[0;350]$. And adding to each of these elements $100$ would give: $[100;450]$. Hence my question: is it possible to do what I want?



Note that I can settle for the interval $[0;350]$ : in my program, it will be enough if I exclude the numbers present in the interval $[0;99]$.






interval-arithmetic







share|cite|improve this question















I have an interval $[0; 100]$ and would like to map it to this new interval: $[100;350]$.



I thought about multiplying it by $3.5$, but that would give the interval $[0;350]$. And adding to each of these elements $100$ would give: $[100;450]$. Hence my question: is it possible to do what I want?



Note that I can settle for the interval $[0;350]$ : in my program, it will be enough if I exclude the numbers present in the interval $[0;99]$.






interval-arithmetic




interval-arithmetic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 16 mins ago









200_success

667515




667515










asked 19 hours ago









JarsOfJam-Scheduler

1378




1378







  • 7




    how about multiplying by 2.5 instead?
    – Lord Shark the Unknown
    19 hours ago












  • 7




    how about multiplying by 2.5 instead?
    – Lord Shark the Unknown
    19 hours ago







7




7




how about multiplying by 2.5 instead?
– Lord Shark the Unknown
19 hours ago




how about multiplying by 2.5 instead?
– Lord Shark the Unknown
19 hours ago










5 Answers
5






active

oldest

votes

















up vote
11
down vote



accepted










To map from $[a,b]$ to $[c, d]$,



Consider the straight line that connects $(a,c)$ to $(b,d)$.



We have the slope $m = fracd-cb-a,$ we are able to recover $m$.



$$y=mx+C$$



To recover $C$, just substitute one of the value say $(a,c)$ and solve for $C$. For our example, we have $a=0$ and $c=100$.



Hence your transformation can be of the form of $y=mx+100$. Can you compute the $m$ to find what you want?






share|cite|improve this answer



























    up vote
    17
    down vote













    The ratio of the lengths of the intervals is $2.5 :1,$ the position of the left extremity is shifted by $100.$ So take the mapping $$f(x)=2.5x+100.$$






    share|cite|improve this answer





























      up vote
      6
      down vote













      You can consider $xmapsto ax + bcolon [0,100]to [100,350]$ such that $0mapsto 100$ and $100mapsto 350$.



      Thus, $$a cdot 0 + b = 100,\ acdot 100 + b = 350,$$ and solving it gives $a = frac 52$, $b = 100$.



      In general, the same technique works for intervals $[x_1,x_2]$ and $[y_1,y_2]$:



      $$ax_1 + b = y_1\
      ax_2 + b = y_2.$$



      Solving it gives $a = fracy_2 - y_1x_2 -x_1$ and $b = y_1 - ax_1$. All in all, it's a line $$y - y_1 = fracy_2-y_1x_2-x_1(x-x_1).$$ Looks familiar?






      share|cite|improve this answer





























        up vote
        5
        down vote













        Since $100cdot t^0=100$ for any positive $t$, we find $t$ such that $100cdot t^100=350implies t=3.5^0.01$. $$boxedy=100cdot3.5^0.01x$$ In general an exponential mapping from $[a,b]$ to $[c,d]$ is $y=cleft(frac dcright)^fracx-ab-a$.






        share|cite|improve this answer


















        • 1




          Way too complicated; a simple affine transformation works.
          – saulspatz
          13 hours ago






        • 1




          @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
          – TheSimpliFire
          13 hours ago










        • But why the downvote?
          – TheSimpliFire
          12 hours ago






        • 1




          I don't think the answer contributes anything positive to the discussion.
          – saulspatz
          12 hours ago






        • 4




          @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
          – TheSimpliFire
          12 hours ago

















        up vote
        2
        down vote













        Another method that’s a bit more general and will come in handy if you want to map arbitrary curves is to parameterize your paths. That is, find a one-to-one mapping from your first path to $[0,1]$, $(0,1]$, or so on as appropriate. Then find a one-to-one mapping from $[0,1]$ to your second path. (The interval $[0,1]$ isn’t special, just convenient.) Finally, compose them.



        Let’s say you want to map $x^2$ over the interval $[0,4]$ to $sin x$ over the interval $[0,2pi]$. A one-to-one mapping from the parabola to the line segment, $t: [0,4] to [0,1]$, is $t = sqrtx/2$, and a mapping from $t in [0,1]$ to a sine wave over $[0,2pi]$ is $sin 2pi t$. Substituting, we get $sin left( pi sqrtxright)$.






        share|cite|improve this answer






















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          5 Answers
          5






          active

          oldest

          votes








          5 Answers
          5






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          11
          down vote



          accepted










          To map from $[a,b]$ to $[c, d]$,



          Consider the straight line that connects $(a,c)$ to $(b,d)$.



          We have the slope $m = fracd-cb-a,$ we are able to recover $m$.



          $$y=mx+C$$



          To recover $C$, just substitute one of the value say $(a,c)$ and solve for $C$. For our example, we have $a=0$ and $c=100$.



          Hence your transformation can be of the form of $y=mx+100$. Can you compute the $m$ to find what you want?






          share|cite|improve this answer
























            up vote
            11
            down vote



            accepted










            To map from $[a,b]$ to $[c, d]$,



            Consider the straight line that connects $(a,c)$ to $(b,d)$.



            We have the slope $m = fracd-cb-a,$ we are able to recover $m$.



            $$y=mx+C$$



            To recover $C$, just substitute one of the value say $(a,c)$ and solve for $C$. For our example, we have $a=0$ and $c=100$.



            Hence your transformation can be of the form of $y=mx+100$. Can you compute the $m$ to find what you want?






            share|cite|improve this answer






















              up vote
              11
              down vote



              accepted







              up vote
              11
              down vote



              accepted






              To map from $[a,b]$ to $[c, d]$,



              Consider the straight line that connects $(a,c)$ to $(b,d)$.



              We have the slope $m = fracd-cb-a,$ we are able to recover $m$.



              $$y=mx+C$$



              To recover $C$, just substitute one of the value say $(a,c)$ and solve for $C$. For our example, we have $a=0$ and $c=100$.



              Hence your transformation can be of the form of $y=mx+100$. Can you compute the $m$ to find what you want?






              share|cite|improve this answer












              To map from $[a,b]$ to $[c, d]$,



              Consider the straight line that connects $(a,c)$ to $(b,d)$.



              We have the slope $m = fracd-cb-a,$ we are able to recover $m$.



              $$y=mx+C$$



              To recover $C$, just substitute one of the value say $(a,c)$ and solve for $C$. For our example, we have $a=0$ and $c=100$.



              Hence your transformation can be of the form of $y=mx+100$. Can you compute the $m$ to find what you want?







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered 19 hours ago









              Siong Thye Goh

              86.2k1459108




              86.2k1459108




















                  up vote
                  17
                  down vote













                  The ratio of the lengths of the intervals is $2.5 :1,$ the position of the left extremity is shifted by $100.$ So take the mapping $$f(x)=2.5x+100.$$






                  share|cite|improve this answer


























                    up vote
                    17
                    down vote













                    The ratio of the lengths of the intervals is $2.5 :1,$ the position of the left extremity is shifted by $100.$ So take the mapping $$f(x)=2.5x+100.$$






                    share|cite|improve this answer
























                      up vote
                      17
                      down vote










                      up vote
                      17
                      down vote









                      The ratio of the lengths of the intervals is $2.5 :1,$ the position of the left extremity is shifted by $100.$ So take the mapping $$f(x)=2.5x+100.$$






                      share|cite|improve this answer














                      The ratio of the lengths of the intervals is $2.5 :1,$ the position of the left extremity is shifted by $100.$ So take the mapping $$f(x)=2.5x+100.$$







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited 13 hours ago









                      Jaideep Khare

                      17.6k32467




                      17.6k32467










                      answered 19 hours ago









                      user376343

                      1,542615




                      1,542615




















                          up vote
                          6
                          down vote













                          You can consider $xmapsto ax + bcolon [0,100]to [100,350]$ such that $0mapsto 100$ and $100mapsto 350$.



                          Thus, $$a cdot 0 + b = 100,\ acdot 100 + b = 350,$$ and solving it gives $a = frac 52$, $b = 100$.



                          In general, the same technique works for intervals $[x_1,x_2]$ and $[y_1,y_2]$:



                          $$ax_1 + b = y_1\
                          ax_2 + b = y_2.$$



                          Solving it gives $a = fracy_2 - y_1x_2 -x_1$ and $b = y_1 - ax_1$. All in all, it's a line $$y - y_1 = fracy_2-y_1x_2-x_1(x-x_1).$$ Looks familiar?






                          share|cite|improve this answer


























                            up vote
                            6
                            down vote













                            You can consider $xmapsto ax + bcolon [0,100]to [100,350]$ such that $0mapsto 100$ and $100mapsto 350$.



                            Thus, $$a cdot 0 + b = 100,\ acdot 100 + b = 350,$$ and solving it gives $a = frac 52$, $b = 100$.



                            In general, the same technique works for intervals $[x_1,x_2]$ and $[y_1,y_2]$:



                            $$ax_1 + b = y_1\
                            ax_2 + b = y_2.$$



                            Solving it gives $a = fracy_2 - y_1x_2 -x_1$ and $b = y_1 - ax_1$. All in all, it's a line $$y - y_1 = fracy_2-y_1x_2-x_1(x-x_1).$$ Looks familiar?






                            share|cite|improve this answer
























                              up vote
                              6
                              down vote










                              up vote
                              6
                              down vote









                              You can consider $xmapsto ax + bcolon [0,100]to [100,350]$ such that $0mapsto 100$ and $100mapsto 350$.



                              Thus, $$a cdot 0 + b = 100,\ acdot 100 + b = 350,$$ and solving it gives $a = frac 52$, $b = 100$.



                              In general, the same technique works for intervals $[x_1,x_2]$ and $[y_1,y_2]$:



                              $$ax_1 + b = y_1\
                              ax_2 + b = y_2.$$



                              Solving it gives $a = fracy_2 - y_1x_2 -x_1$ and $b = y_1 - ax_1$. All in all, it's a line $$y - y_1 = fracy_2-y_1x_2-x_1(x-x_1).$$ Looks familiar?






                              share|cite|improve this answer














                              You can consider $xmapsto ax + bcolon [0,100]to [100,350]$ such that $0mapsto 100$ and $100mapsto 350$.



                              Thus, $$a cdot 0 + b = 100,\ acdot 100 + b = 350,$$ and solving it gives $a = frac 52$, $b = 100$.



                              In general, the same technique works for intervals $[x_1,x_2]$ and $[y_1,y_2]$:



                              $$ax_1 + b = y_1\
                              ax_2 + b = y_2.$$



                              Solving it gives $a = fracy_2 - y_1x_2 -x_1$ and $b = y_1 - ax_1$. All in all, it's a line $$y - y_1 = fracy_2-y_1x_2-x_1(x-x_1).$$ Looks familiar?







                              share|cite|improve this answer














                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited 19 hours ago

























                              answered 19 hours ago









                              Ennar

                              13.4k32343




                              13.4k32343




















                                  up vote
                                  5
                                  down vote













                                  Since $100cdot t^0=100$ for any positive $t$, we find $t$ such that $100cdot t^100=350implies t=3.5^0.01$. $$boxedy=100cdot3.5^0.01x$$ In general an exponential mapping from $[a,b]$ to $[c,d]$ is $y=cleft(frac dcright)^fracx-ab-a$.






                                  share|cite|improve this answer


















                                  • 1




                                    Way too complicated; a simple affine transformation works.
                                    – saulspatz
                                    13 hours ago






                                  • 1




                                    @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
                                    – TheSimpliFire
                                    13 hours ago










                                  • But why the downvote?
                                    – TheSimpliFire
                                    12 hours ago






                                  • 1




                                    I don't think the answer contributes anything positive to the discussion.
                                    – saulspatz
                                    12 hours ago






                                  • 4




                                    @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
                                    – TheSimpliFire
                                    12 hours ago














                                  up vote
                                  5
                                  down vote













                                  Since $100cdot t^0=100$ for any positive $t$, we find $t$ such that $100cdot t^100=350implies t=3.5^0.01$. $$boxedy=100cdot3.5^0.01x$$ In general an exponential mapping from $[a,b]$ to $[c,d]$ is $y=cleft(frac dcright)^fracx-ab-a$.






                                  share|cite|improve this answer


















                                  • 1




                                    Way too complicated; a simple affine transformation works.
                                    – saulspatz
                                    13 hours ago






                                  • 1




                                    @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
                                    – TheSimpliFire
                                    13 hours ago










                                  • But why the downvote?
                                    – TheSimpliFire
                                    12 hours ago






                                  • 1




                                    I don't think the answer contributes anything positive to the discussion.
                                    – saulspatz
                                    12 hours ago






                                  • 4




                                    @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
                                    – TheSimpliFire
                                    12 hours ago












                                  up vote
                                  5
                                  down vote










                                  up vote
                                  5
                                  down vote









                                  Since $100cdot t^0=100$ for any positive $t$, we find $t$ such that $100cdot t^100=350implies t=3.5^0.01$. $$boxedy=100cdot3.5^0.01x$$ In general an exponential mapping from $[a,b]$ to $[c,d]$ is $y=cleft(frac dcright)^fracx-ab-a$.






                                  share|cite|improve this answer














                                  Since $100cdot t^0=100$ for any positive $t$, we find $t$ such that $100cdot t^100=350implies t=3.5^0.01$. $$boxedy=100cdot3.5^0.01x$$ In general an exponential mapping from $[a,b]$ to $[c,d]$ is $y=cleft(frac dcright)^fracx-ab-a$.







                                  share|cite|improve this answer














                                  share|cite|improve this answer



                                  share|cite|improve this answer








                                  edited 15 hours ago

























                                  answered 15 hours ago









                                  TheSimpliFire

                                  11.4k62256




                                  11.4k62256







                                  • 1




                                    Way too complicated; a simple affine transformation works.
                                    – saulspatz
                                    13 hours ago






                                  • 1




                                    @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
                                    – TheSimpliFire
                                    13 hours ago










                                  • But why the downvote?
                                    – TheSimpliFire
                                    12 hours ago






                                  • 1




                                    I don't think the answer contributes anything positive to the discussion.
                                    – saulspatz
                                    12 hours ago






                                  • 4




                                    @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
                                    – TheSimpliFire
                                    12 hours ago












                                  • 1




                                    Way too complicated; a simple affine transformation works.
                                    – saulspatz
                                    13 hours ago






                                  • 1




                                    @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
                                    – TheSimpliFire
                                    13 hours ago










                                  • But why the downvote?
                                    – TheSimpliFire
                                    12 hours ago






                                  • 1




                                    I don't think the answer contributes anything positive to the discussion.
                                    – saulspatz
                                    12 hours ago






                                  • 4




                                    @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
                                    – TheSimpliFire
                                    12 hours ago







                                  1




                                  1




                                  Way too complicated; a simple affine transformation works.
                                  – saulspatz
                                  13 hours ago




                                  Way too complicated; a simple affine transformation works.
                                  – saulspatz
                                  13 hours ago




                                  1




                                  1




                                  @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
                                  – TheSimpliFire
                                  13 hours ago




                                  @saulspatz I know, since the linear transformation has already been said three times. I just wanted to give the next best approach.
                                  – TheSimpliFire
                                  13 hours ago












                                  But why the downvote?
                                  – TheSimpliFire
                                  12 hours ago




                                  But why the downvote?
                                  – TheSimpliFire
                                  12 hours ago




                                  1




                                  1




                                  I don't think the answer contributes anything positive to the discussion.
                                  – saulspatz
                                  12 hours ago




                                  I don't think the answer contributes anything positive to the discussion.
                                  – saulspatz
                                  12 hours ago




                                  4




                                  4




                                  @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
                                  – TheSimpliFire
                                  12 hours ago




                                  @saulspatz My answer is an answer to the question: 'Is it possible to transform an interval into another?' What the OP specifically wanted has already been discussed many times, and I do not see any harm in adding an alternative method to perform such a transformation.
                                  – TheSimpliFire
                                  12 hours ago










                                  up vote
                                  2
                                  down vote













                                  Another method that’s a bit more general and will come in handy if you want to map arbitrary curves is to parameterize your paths. That is, find a one-to-one mapping from your first path to $[0,1]$, $(0,1]$, or so on as appropriate. Then find a one-to-one mapping from $[0,1]$ to your second path. (The interval $[0,1]$ isn’t special, just convenient.) Finally, compose them.



                                  Let’s say you want to map $x^2$ over the interval $[0,4]$ to $sin x$ over the interval $[0,2pi]$. A one-to-one mapping from the parabola to the line segment, $t: [0,4] to [0,1]$, is $t = sqrtx/2$, and a mapping from $t in [0,1]$ to a sine wave over $[0,2pi]$ is $sin 2pi t$. Substituting, we get $sin left( pi sqrtxright)$.






                                  share|cite|improve this answer


























                                    up vote
                                    2
                                    down vote













                                    Another method that’s a bit more general and will come in handy if you want to map arbitrary curves is to parameterize your paths. That is, find a one-to-one mapping from your first path to $[0,1]$, $(0,1]$, or so on as appropriate. Then find a one-to-one mapping from $[0,1]$ to your second path. (The interval $[0,1]$ isn’t special, just convenient.) Finally, compose them.



                                    Let’s say you want to map $x^2$ over the interval $[0,4]$ to $sin x$ over the interval $[0,2pi]$. A one-to-one mapping from the parabola to the line segment, $t: [0,4] to [0,1]$, is $t = sqrtx/2$, and a mapping from $t in [0,1]$ to a sine wave over $[0,2pi]$ is $sin 2pi t$. Substituting, we get $sin left( pi sqrtxright)$.






                                    share|cite|improve this answer
























                                      up vote
                                      2
                                      down vote










                                      up vote
                                      2
                                      down vote









                                      Another method that’s a bit more general and will come in handy if you want to map arbitrary curves is to parameterize your paths. That is, find a one-to-one mapping from your first path to $[0,1]$, $(0,1]$, or so on as appropriate. Then find a one-to-one mapping from $[0,1]$ to your second path. (The interval $[0,1]$ isn’t special, just convenient.) Finally, compose them.



                                      Let’s say you want to map $x^2$ over the interval $[0,4]$ to $sin x$ over the interval $[0,2pi]$. A one-to-one mapping from the parabola to the line segment, $t: [0,4] to [0,1]$, is $t = sqrtx/2$, and a mapping from $t in [0,1]$ to a sine wave over $[0,2pi]$ is $sin 2pi t$. Substituting, we get $sin left( pi sqrtxright)$.






                                      share|cite|improve this answer














                                      Another method that’s a bit more general and will come in handy if you want to map arbitrary curves is to parameterize your paths. That is, find a one-to-one mapping from your first path to $[0,1]$, $(0,1]$, or so on as appropriate. Then find a one-to-one mapping from $[0,1]$ to your second path. (The interval $[0,1]$ isn’t special, just convenient.) Finally, compose them.



                                      Let’s say you want to map $x^2$ over the interval $[0,4]$ to $sin x$ over the interval $[0,2pi]$. A one-to-one mapping from the parabola to the line segment, $t: [0,4] to [0,1]$, is $t = sqrtx/2$, and a mapping from $t in [0,1]$ to a sine wave over $[0,2pi]$ is $sin 2pi t$. Substituting, we get $sin left( pi sqrtxright)$.







                                      share|cite|improve this answer














                                      share|cite|improve this answer



                                      share|cite|improve this answer








                                      edited 10 hours ago

























                                      answered 10 hours ago









                                      Davislor

                                      2,260715




                                      2,260715



























                                           

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