Largest Orthogonal circle for a given circle

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What is the radius of the largest orthogonal circle (as a function of R)that can be constructed for a given circle of radius R?



Definition of Largest circle-the circle that spans the longest section of the circumference of given circle of radius R on it’s convex side




geometry euclidean-geometry circle orthogonality mathematica




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  • 1




    no upper bound.
    – Will Jagy
    Jul 28 at 3:26










  • @WillJagy what if I change the definition of the”largest circle” to the circle that spans the longest section of the circumference on it’s convex side
    – user472374
    Jul 28 at 3:27










  • @WillJagy i can’t wrap my head around NO UPPER BOUND, rather intuitively, I speculate NO LOWER BOUND
    – user472374
    Jul 28 at 3:37










  • @user472374: The "largest orthogonal circle" is a straight line through the center. For any other orthogonal circle-circle you may have, there's a larger orthogonal circle-circle (which has less curvature and is therefore closer to being a straight line). This is what WillJagy means by "no upper bound".
    – Blue
    Jul 28 at 3:37











  • @Blue Yeah, I get it now
    – user472374
    Jul 28 at 4:30














up vote
1
down vote

favorite












What is the radius of the largest orthogonal circle (as a function of R)that can be constructed for a given circle of radius R?



Definition of Largest circle-the circle that spans the longest section of the circumference of given circle of radius R on it’s convex side




geometry euclidean-geometry circle orthogonality mathematica




share|cite|improve this question

















  • 1




    no upper bound.
    – Will Jagy
    Jul 28 at 3:26










  • @WillJagy what if I change the definition of the”largest circle” to the circle that spans the longest section of the circumference on it’s convex side
    – user472374
    Jul 28 at 3:27










  • @WillJagy i can’t wrap my head around NO UPPER BOUND, rather intuitively, I speculate NO LOWER BOUND
    – user472374
    Jul 28 at 3:37










  • @user472374: The "largest orthogonal circle" is a straight line through the center. For any other orthogonal circle-circle you may have, there's a larger orthogonal circle-circle (which has less curvature and is therefore closer to being a straight line). This is what WillJagy means by "no upper bound".
    – Blue
    Jul 28 at 3:37











  • @Blue Yeah, I get it now
    – user472374
    Jul 28 at 4:30












up vote
1
down vote

favorite









up vote
1
down vote

favorite











What is the radius of the largest orthogonal circle (as a function of R)that can be constructed for a given circle of radius R?



Definition of Largest circle-the circle that spans the longest section of the circumference of given circle of radius R on it’s convex side




geometry euclidean-geometry circle orthogonality mathematica




share|cite|improve this question













What is the radius of the largest orthogonal circle (as a function of R)that can be constructed for a given circle of radius R?



Definition of Largest circle-the circle that spans the longest section of the circumference of given circle of radius R on it’s convex side





geometry euclidean-geometry circle orthogonality mathematica





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 28 at 3:29
























asked Jul 28 at 3:23









user472374

63




63







  • 1




    no upper bound.
    – Will Jagy
    Jul 28 at 3:26










  • @WillJagy what if I change the definition of the”largest circle” to the circle that spans the longest section of the circumference on it’s convex side
    – user472374
    Jul 28 at 3:27










  • @WillJagy i can’t wrap my head around NO UPPER BOUND, rather intuitively, I speculate NO LOWER BOUND
    – user472374
    Jul 28 at 3:37










  • @user472374: The "largest orthogonal circle" is a straight line through the center. For any other orthogonal circle-circle you may have, there's a larger orthogonal circle-circle (which has less curvature and is therefore closer to being a straight line). This is what WillJagy means by "no upper bound".
    – Blue
    Jul 28 at 3:37











  • @Blue Yeah, I get it now
    – user472374
    Jul 28 at 4:30












  • 1




    no upper bound.
    – Will Jagy
    Jul 28 at 3:26










  • @WillJagy what if I change the definition of the”largest circle” to the circle that spans the longest section of the circumference on it’s convex side
    – user472374
    Jul 28 at 3:27










  • @WillJagy i can’t wrap my head around NO UPPER BOUND, rather intuitively, I speculate NO LOWER BOUND
    – user472374
    Jul 28 at 3:37










  • @user472374: The "largest orthogonal circle" is a straight line through the center. For any other orthogonal circle-circle you may have, there's a larger orthogonal circle-circle (which has less curvature and is therefore closer to being a straight line). This is what WillJagy means by "no upper bound".
    – Blue
    Jul 28 at 3:37











  • @Blue Yeah, I get it now
    – user472374
    Jul 28 at 4:30







1




1




no upper bound.
– Will Jagy
Jul 28 at 3:26




no upper bound.
– Will Jagy
Jul 28 at 3:26












@WillJagy what if I change the definition of the”largest circle” to the circle that spans the longest section of the circumference on it’s convex side
– user472374
Jul 28 at 3:27




@WillJagy what if I change the definition of the”largest circle” to the circle that spans the longest section of the circumference on it’s convex side
– user472374
Jul 28 at 3:27












@WillJagy i can’t wrap my head around NO UPPER BOUND, rather intuitively, I speculate NO LOWER BOUND
– user472374
Jul 28 at 3:37




@WillJagy i can’t wrap my head around NO UPPER BOUND, rather intuitively, I speculate NO LOWER BOUND
– user472374
Jul 28 at 3:37












@user472374: The "largest orthogonal circle" is a straight line through the center. For any other orthogonal circle-circle you may have, there's a larger orthogonal circle-circle (which has less curvature and is therefore closer to being a straight line). This is what WillJagy means by "no upper bound".
– Blue
Jul 28 at 3:37





@user472374: The "largest orthogonal circle" is a straight line through the center. For any other orthogonal circle-circle you may have, there's a larger orthogonal circle-circle (which has less curvature and is therefore closer to being a straight line). This is what WillJagy means by "no upper bound".
– Blue
Jul 28 at 3:37













@Blue Yeah, I get it now
– user472374
Jul 28 at 4:30




@Blue Yeah, I get it now
– user472374
Jul 28 at 4:30










1 Answer
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enter image description here



For any (true) circle orthogonal to a given circle, you can always find a larger one, with ever-more of the given circle's circumference on its "convex side". So, there is no "largest" (true) circle.



Now, larger and larger circles have smaller and smaller curvature; larger and larger orthogonal circles get closer and closer to the center of the given circle. Thus, this sequence of circles, as they say, "tend toward" a straight line through the given circle's center. It's sometimes philosophically helpful to consider a line to be a circle, just one of infinite radius. (A number of theorems become easier to state.) In that sense, the "largest circle" is that line through the center.






share|cite|improve this answer

















  • 1




    Thanks You for your answer!
    – user472374
    Jul 28 at 4:36










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1
down vote













enter image description here



For any (true) circle orthogonal to a given circle, you can always find a larger one, with ever-more of the given circle's circumference on its "convex side". So, there is no "largest" (true) circle.



Now, larger and larger circles have smaller and smaller curvature; larger and larger orthogonal circles get closer and closer to the center of the given circle. Thus, this sequence of circles, as they say, "tend toward" a straight line through the given circle's center. It's sometimes philosophically helpful to consider a line to be a circle, just one of infinite radius. (A number of theorems become easier to state.) In that sense, the "largest circle" is that line through the center.






share|cite|improve this answer

















  • 1




    Thanks You for your answer!
    – user472374
    Jul 28 at 4:36














up vote
1
down vote













enter image description here



For any (true) circle orthogonal to a given circle, you can always find a larger one, with ever-more of the given circle's circumference on its "convex side". So, there is no "largest" (true) circle.



Now, larger and larger circles have smaller and smaller curvature; larger and larger orthogonal circles get closer and closer to the center of the given circle. Thus, this sequence of circles, as they say, "tend toward" a straight line through the given circle's center. It's sometimes philosophically helpful to consider a line to be a circle, just one of infinite radius. (A number of theorems become easier to state.) In that sense, the "largest circle" is that line through the center.






share|cite|improve this answer

















  • 1




    Thanks You for your answer!
    – user472374
    Jul 28 at 4:36












up vote
1
down vote










up vote
1
down vote









enter image description here



For any (true) circle orthogonal to a given circle, you can always find a larger one, with ever-more of the given circle's circumference on its "convex side". So, there is no "largest" (true) circle.



Now, larger and larger circles have smaller and smaller curvature; larger and larger orthogonal circles get closer and closer to the center of the given circle. Thus, this sequence of circles, as they say, "tend toward" a straight line through the given circle's center. It's sometimes philosophically helpful to consider a line to be a circle, just one of infinite radius. (A number of theorems become easier to state.) In that sense, the "largest circle" is that line through the center.






share|cite|improve this answer













enter image description here



For any (true) circle orthogonal to a given circle, you can always find a larger one, with ever-more of the given circle's circumference on its "convex side". So, there is no "largest" (true) circle.



Now, larger and larger circles have smaller and smaller curvature; larger and larger orthogonal circles get closer and closer to the center of the given circle. Thus, this sequence of circles, as they say, "tend toward" a straight line through the given circle's center. It's sometimes philosophically helpful to consider a line to be a circle, just one of infinite radius. (A number of theorems become easier to state.) In that sense, the "largest circle" is that line through the center.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 28 at 4:35









Blue

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  • 1




    Thanks You for your answer!
    – user472374
    Jul 28 at 4:36












  • 1




    Thanks You for your answer!
    – user472374
    Jul 28 at 4:36







1




1




Thanks You for your answer!
– user472374
Jul 28 at 4:36




Thanks You for your answer!
– user472374
Jul 28 at 4:36












 

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