Circle whose radius is infinite
I have the intuition that a circle whose radius is infinite is a straight line.
Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision.
What I would faithfully ask for is a more accurate expression of this idea ,if it were true, in addition to a proof thereof.
I would appreciate it if I were provided ,besides your own answers, with some sources to refer to.
Many thanks for considering my request.
geometry differential-geometry circle hyperbolic-geometry inversive-geometry
|
show 2 more comments
I have the intuition that a circle whose radius is infinite is a straight line.
Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision.
What I would faithfully ask for is a more accurate expression of this idea ,if it were true, in addition to a proof thereof.
I would appreciate it if I were provided ,besides your own answers, with some sources to refer to.
Many thanks for considering my request.
geometry differential-geometry circle hyperbolic-geometry inversive-geometry
2
There is no such thing as a circle with infinite radius.
– Ben W
2 days ago
5
There is such a thing as the limit of a circle whose radius tends to infinity and it passes through two fixed points, and that limit is the line passing through the fixed points.
– Oscar Lanzi
2 days ago
1
en.wikipedia.org/wiki/Horocycle and then en.wikipedia.org/wiki/Hypercycle_(geometry)
– Will Jagy
2 days ago
1
There are many situations in which thinking of a line as a circle of infinite radius is helpful. See, for instance, Descartes' Theorem relating the curvatures of four mutually-tangent circles. The formulas "just work" when one or more of the circles is a line, and the corresponding curvature is $0$. (Note that curvature is the reciprocal of radius, so "zero curvature" matches "infinite radius".)
– Blue
2 days ago
1
@BenW: In geometry with the euclidean plane extended to the projective plane, it is common to talk about the line at infinity, which is also a circle with infinite radius. Any non-degenerate circle with centre on the line at infinity is a line in the plane. In the projective plane, any two distinct points have a unique line through them, and any two distinct lines have a unique intersection, and any two distinct intersecting circles have exactly two intersections counting multiplicity. Blue gave an example of a formula involving radius that requires a line to have infinite radius.
– user21820
2 days ago
|
show 2 more comments
I have the intuition that a circle whose radius is infinite is a straight line.
Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision.
What I would faithfully ask for is a more accurate expression of this idea ,if it were true, in addition to a proof thereof.
I would appreciate it if I were provided ,besides your own answers, with some sources to refer to.
Many thanks for considering my request.
geometry differential-geometry circle hyperbolic-geometry inversive-geometry
I have the intuition that a circle whose radius is infinite is a straight line.
Nonetheless, I don’t feel that what I’ve just stated is really scientific as it has some vagueness and lacks precision.
What I would faithfully ask for is a more accurate expression of this idea ,if it were true, in addition to a proof thereof.
I would appreciate it if I were provided ,besides your own answers, with some sources to refer to.
Many thanks for considering my request.
geometry differential-geometry circle hyperbolic-geometry inversive-geometry
geometry differential-geometry circle hyperbolic-geometry inversive-geometry
edited 2 days ago
asked 2 days ago
user630906
2
There is no such thing as a circle with infinite radius.
– Ben W
2 days ago
5
There is such a thing as the limit of a circle whose radius tends to infinity and it passes through two fixed points, and that limit is the line passing through the fixed points.
– Oscar Lanzi
2 days ago
1
en.wikipedia.org/wiki/Horocycle and then en.wikipedia.org/wiki/Hypercycle_(geometry)
– Will Jagy
2 days ago
1
There are many situations in which thinking of a line as a circle of infinite radius is helpful. See, for instance, Descartes' Theorem relating the curvatures of four mutually-tangent circles. The formulas "just work" when one or more of the circles is a line, and the corresponding curvature is $0$. (Note that curvature is the reciprocal of radius, so "zero curvature" matches "infinite radius".)
– Blue
2 days ago
1
@BenW: In geometry with the euclidean plane extended to the projective plane, it is common to talk about the line at infinity, which is also a circle with infinite radius. Any non-degenerate circle with centre on the line at infinity is a line in the plane. In the projective plane, any two distinct points have a unique line through them, and any two distinct lines have a unique intersection, and any two distinct intersecting circles have exactly two intersections counting multiplicity. Blue gave an example of a formula involving radius that requires a line to have infinite radius.
– user21820
2 days ago
|
show 2 more comments
2
There is no such thing as a circle with infinite radius.
– Ben W
2 days ago
5
There is such a thing as the limit of a circle whose radius tends to infinity and it passes through two fixed points, and that limit is the line passing through the fixed points.
– Oscar Lanzi
2 days ago
1
en.wikipedia.org/wiki/Horocycle and then en.wikipedia.org/wiki/Hypercycle_(geometry)
– Will Jagy
2 days ago
1
There are many situations in which thinking of a line as a circle of infinite radius is helpful. See, for instance, Descartes' Theorem relating the curvatures of four mutually-tangent circles. The formulas "just work" when one or more of the circles is a line, and the corresponding curvature is $0$. (Note that curvature is the reciprocal of radius, so "zero curvature" matches "infinite radius".)
– Blue
2 days ago
1
@BenW: In geometry with the euclidean plane extended to the projective plane, it is common to talk about the line at infinity, which is also a circle with infinite radius. Any non-degenerate circle with centre on the line at infinity is a line in the plane. In the projective plane, any two distinct points have a unique line through them, and any two distinct lines have a unique intersection, and any two distinct intersecting circles have exactly two intersections counting multiplicity. Blue gave an example of a formula involving radius that requires a line to have infinite radius.
– user21820
2 days ago
2
2
There is no such thing as a circle with infinite radius.
– Ben W
2 days ago
There is no such thing as a circle with infinite radius.
– Ben W
2 days ago
5
5
There is such a thing as the limit of a circle whose radius tends to infinity and it passes through two fixed points, and that limit is the line passing through the fixed points.
– Oscar Lanzi
2 days ago
There is such a thing as the limit of a circle whose radius tends to infinity and it passes through two fixed points, and that limit is the line passing through the fixed points.
– Oscar Lanzi
2 days ago
1
1
en.wikipedia.org/wiki/Horocycle and then en.wikipedia.org/wiki/Hypercycle_(geometry)
– Will Jagy
2 days ago
en.wikipedia.org/wiki/Horocycle and then en.wikipedia.org/wiki/Hypercycle_(geometry)
– Will Jagy
2 days ago
1
1
There are many situations in which thinking of a line as a circle of infinite radius is helpful. See, for instance, Descartes' Theorem relating the curvatures of four mutually-tangent circles. The formulas "just work" when one or more of the circles is a line, and the corresponding curvature is $0$. (Note that curvature is the reciprocal of radius, so "zero curvature" matches "infinite radius".)
– Blue
2 days ago
There are many situations in which thinking of a line as a circle of infinite radius is helpful. See, for instance, Descartes' Theorem relating the curvatures of four mutually-tangent circles. The formulas "just work" when one or more of the circles is a line, and the corresponding curvature is $0$. (Note that curvature is the reciprocal of radius, so "zero curvature" matches "infinite radius".)
– Blue
2 days ago
1
1
@BenW: In geometry with the euclidean plane extended to the projective plane, it is common to talk about the line at infinity, which is also a circle with infinite radius. Any non-degenerate circle with centre on the line at infinity is a line in the plane. In the projective plane, any two distinct points have a unique line through them, and any two distinct lines have a unique intersection, and any two distinct intersecting circles have exactly two intersections counting multiplicity. Blue gave an example of a formula involving radius that requires a line to have infinite radius.
– user21820
2 days ago
@BenW: In geometry with the euclidean plane extended to the projective plane, it is common to talk about the line at infinity, which is also a circle with infinite radius. Any non-degenerate circle with centre on the line at infinity is a line in the plane. In the projective plane, any two distinct points have a unique line through them, and any two distinct lines have a unique intersection, and any two distinct intersecting circles have exactly two intersections counting multiplicity. Blue gave an example of a formula involving radius that requires a line to have infinite radius.
– user21820
2 days ago
|
show 2 more comments
3 Answers
3
active
oldest
votes
A "circle", by definition, has finite radius. One can say that the limit of a sequence of circles, all passing through two given points, is the straight line passing through those two points.
answered 2 days ago
user247327
10.4k1515
add a comment |
The answer you are looking for kind of depends, as your question is not truely well-defined.
The answer of @user247327, assuming additionally 2 given points and describing a limiting process, which focusses on those provided points, might serve as one possibility.
But there are answers too, which do focus onto the circle center instead.
Eg. one such answer you'd get, when considering circles around the origin with increasing radii within the real projective plane. Then too you'd get the "line of infinity" as the according limit.
A different answer would be obtained, when considering the one-point compactification of $mathbb{C}=mathbb{R}^2$. There circles around the origin with ever increasing radii finally result in that additional single point of infinity only. That one then is nothing but the opposite point to the origin on the Riemann sphere.
(For according references just google after the provided terms.)
--- rk
answered 2 days ago
Dr. Richard Klitzing
1,3516
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
add a comment |
Suppose you take a straight line $L$ and fix a point $P in L$. Consider a variable point $C in L$, and consider the circle with center $C$ and radius $CP$. This circle crosses the line at right angles at the point $P$. As $C$ goes to infinity in one direction along the line, what can happen? Is there some limiting shape to this circle as $C$ approaches an infinite end of the line $L$?
If you were doing this in the Euclidean plane, then the answer is given by @user247327, the limiting shape is the straight line through $P$ perpendicular to $L$.
However, I notice that you have the tag hyperbolic-geometry on your question. In the hyperbolic plane, the answer is different: the limit is a horocycle, a curve of constant and nonzero curvature which does not close up to form a circle, but the two endpoints of that horocycle instead converge (in the compactification of the hyperbolic plane with its circle at infinity) to the same ideal endpoint of $L$ that $C$ is approaching.
answered 9 hours ago
Lee Mosher
48.2k33681
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058712%2fcircle-whose-radius-is-infinite%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
A "circle", by definition, has finite radius. One can say that the limit of a sequence of circles, all passing through two given points, is the straight line passing through those two points.
answered 2 days ago
user247327
10.4k1515
add a comment |
A "circle", by definition, has finite radius. One can say that the limit of a sequence of circles, all passing through two given points, is the straight line passing through those two points.
answered 2 days ago
user247327
10.4k1515
add a comment |
A "circle", by definition, has finite radius. One can say that the limit of a sequence of circles, all passing through two given points, is the straight line passing through those two points.
answered 2 days ago
user247327
10.4k1515
A "circle", by definition, has finite radius. One can say that the limit of a sequence of circles, all passing through two given points, is the straight line passing through those two points.
answered 2 days ago
user247327
10.4k1515
answered 2 days ago
user247327
10.4k1515
answered 2 days ago
user247327
10.4k1515
answered 2 days ago
user247327
10.4k1515
10.4k1515
add a comment |
add a comment |
The answer you are looking for kind of depends, as your question is not truely well-defined.
The answer of @user247327, assuming additionally 2 given points and describing a limiting process, which focusses on those provided points, might serve as one possibility.
But there are answers too, which do focus onto the circle center instead.
Eg. one such answer you'd get, when considering circles around the origin with increasing radii within the real projective plane. Then too you'd get the "line of infinity" as the according limit.
A different answer would be obtained, when considering the one-point compactification of $mathbb{C}=mathbb{R}^2$. There circles around the origin with ever increasing radii finally result in that additional single point of infinity only. That one then is nothing but the opposite point to the origin on the Riemann sphere.
(For according references just google after the provided terms.)
--- rk
answered 2 days ago
Dr. Richard Klitzing
1,3516
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
add a comment |
The answer you are looking for kind of depends, as your question is not truely well-defined.
The answer of @user247327, assuming additionally 2 given points and describing a limiting process, which focusses on those provided points, might serve as one possibility.
But there are answers too, which do focus onto the circle center instead.
Eg. one such answer you'd get, when considering circles around the origin with increasing radii within the real projective plane. Then too you'd get the "line of infinity" as the according limit.
A different answer would be obtained, when considering the one-point compactification of $mathbb{C}=mathbb{R}^2$. There circles around the origin with ever increasing radii finally result in that additional single point of infinity only. That one then is nothing but the opposite point to the origin on the Riemann sphere.
(For according references just google after the provided terms.)
--- rk
answered 2 days ago
Dr. Richard Klitzing
1,3516
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
add a comment |
The answer you are looking for kind of depends, as your question is not truely well-defined.
The answer of @user247327, assuming additionally 2 given points and describing a limiting process, which focusses on those provided points, might serve as one possibility.
But there are answers too, which do focus onto the circle center instead.
Eg. one such answer you'd get, when considering circles around the origin with increasing radii within the real projective plane. Then too you'd get the "line of infinity" as the according limit.
A different answer would be obtained, when considering the one-point compactification of $mathbb{C}=mathbb{R}^2$. There circles around the origin with ever increasing radii finally result in that additional single point of infinity only. That one then is nothing but the opposite point to the origin on the Riemann sphere.
(For according references just google after the provided terms.)
--- rk
answered 2 days ago
Dr. Richard Klitzing
1,3516
The answer you are looking for kind of depends, as your question is not truely well-defined.
The answer of @user247327, assuming additionally 2 given points and describing a limiting process, which focusses on those provided points, might serve as one possibility.
But there are answers too, which do focus onto the circle center instead.
Eg. one such answer you'd get, when considering circles around the origin with increasing radii within the real projective plane. Then too you'd get the "line of infinity" as the according limit.
A different answer would be obtained, when considering the one-point compactification of $mathbb{C}=mathbb{R}^2$. There circles around the origin with ever increasing radii finally result in that additional single point of infinity only. That one then is nothing but the opposite point to the origin on the Riemann sphere.
(For according references just google after the provided terms.)
--- rk
answered 2 days ago
Dr. Richard Klitzing
1,3516
answered 2 days ago
Dr. Richard Klitzing
1,3516
answered 2 days ago
Dr. Richard Klitzing
1,3516
answered 2 days ago
Dr. Richard Klitzing
1,3516
1,3516
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
add a comment |
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
I would add yet another possibility: when considering the family of circles that pass through two fixed points in the plane, it can be useful to consider the line through those points as a member of the family. In the projective plane, its “center” is the point at infinity of the points’ bisector. (Although, to even talk about circles on the projective plane, one must impose a Euclidean geometry on it.)
– amd
2 days ago
add a comment |
Suppose you take a straight line $L$ and fix a point $P in L$. Consider a variable point $C in L$, and consider the circle with center $C$ and radius $CP$. This circle crosses the line at right angles at the point $P$. As $C$ goes to infinity in one direction along the line, what can happen? Is there some limiting shape to this circle as $C$ approaches an infinite end of the line $L$?
If you were doing this in the Euclidean plane, then the answer is given by @user247327, the limiting shape is the straight line through $P$ perpendicular to $L$.
However, I notice that you have the tag hyperbolic-geometry on your question. In the hyperbolic plane, the answer is different: the limit is a horocycle, a curve of constant and nonzero curvature which does not close up to form a circle, but the two endpoints of that horocycle instead converge (in the compactification of the hyperbolic plane with its circle at infinity) to the same ideal endpoint of $L$ that $C$ is approaching.
answered 9 hours ago
Lee Mosher
48.2k33681
add a comment |
Suppose you take a straight line $L$ and fix a point $P in L$. Consider a variable point $C in L$, and consider the circle with center $C$ and radius $CP$. This circle crosses the line at right angles at the point $P$. As $C$ goes to infinity in one direction along the line, what can happen? Is there some limiting shape to this circle as $C$ approaches an infinite end of the line $L$?
If you were doing this in the Euclidean plane, then the answer is given by @user247327, the limiting shape is the straight line through $P$ perpendicular to $L$.
However, I notice that you have the tag hyperbolic-geometry on your question. In the hyperbolic plane, the answer is different: the limit is a horocycle, a curve of constant and nonzero curvature which does not close up to form a circle, but the two endpoints of that horocycle instead converge (in the compactification of the hyperbolic plane with its circle at infinity) to the same ideal endpoint of $L$ that $C$ is approaching.
answered 9 hours ago
Lee Mosher
48.2k33681
add a comment |
Suppose you take a straight line $L$ and fix a point $P in L$. Consider a variable point $C in L$, and consider the circle with center $C$ and radius $CP$. This circle crosses the line at right angles at the point $P$. As $C$ goes to infinity in one direction along the line, what can happen? Is there some limiting shape to this circle as $C$ approaches an infinite end of the line $L$?
If you were doing this in the Euclidean plane, then the answer is given by @user247327, the limiting shape is the straight line through $P$ perpendicular to $L$.
However, I notice that you have the tag hyperbolic-geometry on your question. In the hyperbolic plane, the answer is different: the limit is a horocycle, a curve of constant and nonzero curvature which does not close up to form a circle, but the two endpoints of that horocycle instead converge (in the compactification of the hyperbolic plane with its circle at infinity) to the same ideal endpoint of $L$ that $C$ is approaching.
answered 9 hours ago
Lee Mosher
48.2k33681
Suppose you take a straight line $L$ and fix a point $P in L$. Consider a variable point $C in L$, and consider the circle with center $C$ and radius $CP$. This circle crosses the line at right angles at the point $P$. As $C$ goes to infinity in one direction along the line, what can happen? Is there some limiting shape to this circle as $C$ approaches an infinite end of the line $L$?
If you were doing this in the Euclidean plane, then the answer is given by @user247327, the limiting shape is the straight line through $P$ perpendicular to $L$.
However, I notice that you have the tag hyperbolic-geometry on your question. In the hyperbolic plane, the answer is different: the limit is a horocycle, a curve of constant and nonzero curvature which does not close up to form a circle, but the two endpoints of that horocycle instead converge (in the compactification of the hyperbolic plane with its circle at infinity) to the same ideal endpoint of $L$ that $C$ is approaching.
answered 9 hours ago
Lee Mosher
48.2k33681
edited 9 hours ago
answered 9 hours ago
Lee Mosher
48.2k33681
answered 9 hours ago
Lee Mosher
48.2k33681
answered 9 hours ago
Lee Mosher
48.2k33681
48.2k33681
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3058712%2fcircle-whose-radius-is-infinite%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
There is no such thing as a circle with infinite radius.
– Ben W
2 days ago
5
There is such a thing as the limit of a circle whose radius tends to infinity and it passes through two fixed points, and that limit is the line passing through the fixed points.
– Oscar Lanzi
2 days ago
1
en.wikipedia.org/wiki/Horocycle and then en.wikipedia.org/wiki/Hypercycle_(geometry)
– Will Jagy
2 days ago
1
There are many situations in which thinking of a line as a circle of infinite radius is helpful. See, for instance, Descartes' Theorem relating the curvatures of four mutually-tangent circles. The formulas "just work" when one or more of the circles is a line, and the corresponding curvature is $0$. (Note that curvature is the reciprocal of radius, so "zero curvature" matches "infinite radius".)
– Blue
2 days ago
1
@BenW: In geometry with the euclidean plane extended to the projective plane, it is common to talk about the line at infinity, which is also a circle with infinite radius. Any non-degenerate circle with centre on the line at infinity is a line in the plane. In the projective plane, any two distinct points have a unique line through them, and any two distinct lines have a unique intersection, and any two distinct intersecting circles have exactly two intersections counting multiplicity. Blue gave an example of a formula involving radius that requires a line to have infinite radius.
– user21820
2 days ago