cusp vs. corner? or both?
I searched through books and internet and they all have general definitions of them as follows:
Cusp: where the slope of the tangent line changed from -infinity to +infinity (or the other way around)
Corner: left-sided and right-sided derivatives are different.
And I saw a problem which was asking if there is a corner or a cusp given a graph. The graph looked like:
f(x)=-x, if x<0
=sqrt(x), if x>=0
So in short, one branch was straight, and another branch was curved. I know the point where x=0 is not differentiable. But would it be considered a corner or a cusp? In my opinion, it should be a corner because it does not change from -infinity to +infinity. However, while I was searching, I saw an example of graph that looks similar to that, and the website was calling it a cusp (sorry I cannot find the image anymore).
Also, this is another question, but if a cusp have a slope of either -infinity or +infinity, wouldn't it be a subcategory of vertical tangent?
calculus derivatives
edited Oct 7 '17 at 18:49
rubik
6,72132661
asked Sep 14 '17 at 10:25
Harry Hong
8518
add a comment |
I searched through books and internet and they all have general definitions of them as follows:
Cusp: where the slope of the tangent line changed from -infinity to +infinity (or the other way around)
Corner: left-sided and right-sided derivatives are different.
And I saw a problem which was asking if there is a corner or a cusp given a graph. The graph looked like:
f(x)=-x, if x<0
=sqrt(x), if x>=0
So in short, one branch was straight, and another branch was curved. I know the point where x=0 is not differentiable. But would it be considered a corner or a cusp? In my opinion, it should be a corner because it does not change from -infinity to +infinity. However, while I was searching, I saw an example of graph that looks similar to that, and the website was calling it a cusp (sorry I cannot find the image anymore).
Also, this is another question, but if a cusp have a slope of either -infinity or +infinity, wouldn't it be a subcategory of vertical tangent?
calculus derivatives
edited Oct 7 '17 at 18:49
rubik
6,72132661
asked Sep 14 '17 at 10:25
Harry Hong
8518
1
I think what they meant was that you have a cusp if the two parts are "tangent" to one another. In other words, if the angle they make is $0^circ$. In this case, however, the two curves clearly make a $45^circ$ angle, so I would think it should be called a corner. Still, the definitions aren't entirely clear on what to do in this case.
– Arthur
Sep 14 '17 at 10:31
Thanks! That's what I was thinking. By the definition of the corner, when we have a "regular" cusp, since the one-sided derivatives are different (one is -infinity and another being +infinity), would it be also called a corner?
– Harry Hong
Sep 14 '17 at 10:39
Is it different when it's really undefined? I dunno. To me the definitions are a bit unclear, so I don't know.
– Arthur
Sep 14 '17 at 10:41
add a comment |
I searched through books and internet and they all have general definitions of them as follows:
Cusp: where the slope of the tangent line changed from -infinity to +infinity (or the other way around)
Corner: left-sided and right-sided derivatives are different.
And I saw a problem which was asking if there is a corner or a cusp given a graph. The graph looked like:
f(x)=-x, if x<0
=sqrt(x), if x>=0
So in short, one branch was straight, and another branch was curved. I know the point where x=0 is not differentiable. But would it be considered a corner or a cusp? In my opinion, it should be a corner because it does not change from -infinity to +infinity. However, while I was searching, I saw an example of graph that looks similar to that, and the website was calling it a cusp (sorry I cannot find the image anymore).
Also, this is another question, but if a cusp have a slope of either -infinity or +infinity, wouldn't it be a subcategory of vertical tangent?
calculus derivatives
edited Oct 7 '17 at 18:49
rubik
6,72132661
asked Sep 14 '17 at 10:25
Harry Hong
8518
I searched through books and internet and they all have general definitions of them as follows:
Cusp: where the slope of the tangent line changed from -infinity to +infinity (or the other way around)
Corner: left-sided and right-sided derivatives are different.
And I saw a problem which was asking if there is a corner or a cusp given a graph. The graph looked like:
f(x)=-x, if x<0
=sqrt(x), if x>=0
So in short, one branch was straight, and another branch was curved. I know the point where x=0 is not differentiable. But would it be considered a corner or a cusp? In my opinion, it should be a corner because it does not change from -infinity to +infinity. However, while I was searching, I saw an example of graph that looks similar to that, and the website was calling it a cusp (sorry I cannot find the image anymore).
Also, this is another question, but if a cusp have a slope of either -infinity or +infinity, wouldn't it be a subcategory of vertical tangent?
calculus derivatives
calculus derivatives
edited Oct 7 '17 at 18:49
rubik
6,72132661
asked Sep 14 '17 at 10:25
Harry Hong
8518
edited Oct 7 '17 at 18:49
rubik
6,72132661
asked Sep 14 '17 at 10:25
Harry Hong
8518
edited Oct 7 '17 at 18:49
rubik
6,72132661
edited Oct 7 '17 at 18:49
rubik
6,72132661
edited Oct 7 '17 at 18:49
rubik
6,72132661
6,72132661
asked Sep 14 '17 at 10:25
Harry Hong
8518
asked Sep 14 '17 at 10:25
Harry Hong
8518
asked Sep 14 '17 at 10:25
Harry Hong
8518
8518
1
I think what they meant was that you have a cusp if the two parts are "tangent" to one another. In other words, if the angle they make is $0^circ$. In this case, however, the two curves clearly make a $45^circ$ angle, so I would think it should be called a corner. Still, the definitions aren't entirely clear on what to do in this case.
– Arthur
Sep 14 '17 at 10:31
Thanks! That's what I was thinking. By the definition of the corner, when we have a "regular" cusp, since the one-sided derivatives are different (one is -infinity and another being +infinity), would it be also called a corner?
– Harry Hong
Sep 14 '17 at 10:39
Is it different when it's really undefined? I dunno. To me the definitions are a bit unclear, so I don't know.
– Arthur
Sep 14 '17 at 10:41
add a comment |
1
I think what they meant was that you have a cusp if the two parts are "tangent" to one another. In other words, if the angle they make is $0^circ$. In this case, however, the two curves clearly make a $45^circ$ angle, so I would think it should be called a corner. Still, the definitions aren't entirely clear on what to do in this case.
– Arthur
Sep 14 '17 at 10:31
Thanks! That's what I was thinking. By the definition of the corner, when we have a "regular" cusp, since the one-sided derivatives are different (one is -infinity and another being +infinity), would it be also called a corner?
– Harry Hong
Sep 14 '17 at 10:39
Is it different when it's really undefined? I dunno. To me the definitions are a bit unclear, so I don't know.
– Arthur
Sep 14 '17 at 10:41
1
1
I think what they meant was that you have a cusp if the two parts are "tangent" to one another. In other words, if the angle they make is $0^circ$. In this case, however, the two curves clearly make a $45^circ$ angle, so I would think it should be called a corner. Still, the definitions aren't entirely clear on what to do in this case.
– Arthur
Sep 14 '17 at 10:31
I think what they meant was that you have a cusp if the two parts are "tangent" to one another. In other words, if the angle they make is $0^circ$. In this case, however, the two curves clearly make a $45^circ$ angle, so I would think it should be called a corner. Still, the definitions aren't entirely clear on what to do in this case.
– Arthur
Sep 14 '17 at 10:31
Thanks! That's what I was thinking. By the definition of the corner, when we have a "regular" cusp, since the one-sided derivatives are different (one is -infinity and another being +infinity), would it be also called a corner?
– Harry Hong
Sep 14 '17 at 10:39
Thanks! That's what I was thinking. By the definition of the corner, when we have a "regular" cusp, since the one-sided derivatives are different (one is -infinity and another being +infinity), would it be also called a corner?
– Harry Hong
Sep 14 '17 at 10:39
Is it different when it's really undefined? I dunno. To me the definitions are a bit unclear, so I don't know.
– Arthur
Sep 14 '17 at 10:41
Is it different when it's really undefined? I dunno. To me the definitions are a bit unclear, so I don't know.
– Arthur
Sep 14 '17 at 10:41
add a comment |
2 Answers
2
active
oldest
votes
A corner point has two distinct tangents. A cusp has a single one which is vertical.
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
add a comment |
The first "curve" has DE $ dy/dx= cos y/ cos x,quad y = tan^{-1} (tan x ) $ with slope $+1$ everywhere. A regular continuous curve.
In second curve with a corner it has first degree contact i.e., same $(x,y)$, first and second degree values (slope,curvature) can be different. Here wave equation $ y= cos^{-1} (cos x), $ there is change in slope at the corner point $x=pi$. Slope goes from $1$ to $-1$ via $0$ with tangent parallel to x-axis.
Second example below of curve with a corner DE $ y^{''2} +1- y^2 =0 $ is a periodic curve. BC $ (0,-frac12) $. At the corner $(x approx 1.81,y=1), y^{'}= pm 1.59, y^{''}=0 $.
A cusp has first degree contact, same slope of infinite curvature whose sign changes at cusp location and the slope also passes through either zero or infinity value. Examples are point of contact of a circle when a cycloid type curve is produced by rolling circles on a straight line or another circle. Typically, last point of tracks when direction changes in an automobile in reverse gear.
There are curves that are synthesized from individual curves like discontinuous slope electrical wave-forms (square, trapezium, saw-tooth etc). Its Fourier components are evaluated and next, at discontinuity points slope and $y$ values are averaged out.
As to your last question. A circle of a circular toroid has finite curvature at point of vertical tangency. A cusp has infinite curvature at vertical tangent, but not a sub-category.
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
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%2f2429089%2fcusp-vs-corner-or-both%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
A corner point has two distinct tangents. A cusp has a single one which is vertical.
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
add a comment |
A corner point has two distinct tangents. A cusp has a single one which is vertical.
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
add a comment |
A corner point has two distinct tangents. A cusp has a single one which is vertical.
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
A corner point has two distinct tangents. A cusp has a single one which is vertical.
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
answered Oct 7 '17 at 18:50
Yves Daoust
124k671221
124k671221
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
add a comment |
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
Hello. Please, can you provide a link to a book (or some other source) for definitions? Thanks. If you have a link that would be great.
– user35603
Mar 30 at 16:05
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
@user35603: ask Google.
– Yves Daoust
Mar 30 at 16:33
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
I've asked and did not find any books with proper definitions :(
– user35603
Mar 31 at 12:53
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
@user35603: first hit: en.wikipedia.org/wiki/Cusp_(singularity)
– Yves Daoust
Mar 31 at 14:33
add a comment |
The first "curve" has DE $ dy/dx= cos y/ cos x,quad y = tan^{-1} (tan x ) $ with slope $+1$ everywhere. A regular continuous curve.
In second curve with a corner it has first degree contact i.e., same $(x,y)$, first and second degree values (slope,curvature) can be different. Here wave equation $ y= cos^{-1} (cos x), $ there is change in slope at the corner point $x=pi$. Slope goes from $1$ to $-1$ via $0$ with tangent parallel to x-axis.
Second example below of curve with a corner DE $ y^{''2} +1- y^2 =0 $ is a periodic curve. BC $ (0,-frac12) $. At the corner $(x approx 1.81,y=1), y^{'}= pm 1.59, y^{''}=0 $.
A cusp has first degree contact, same slope of infinite curvature whose sign changes at cusp location and the slope also passes through either zero or infinity value. Examples are point of contact of a circle when a cycloid type curve is produced by rolling circles on a straight line or another circle. Typically, last point of tracks when direction changes in an automobile in reverse gear.
There are curves that are synthesized from individual curves like discontinuous slope electrical wave-forms (square, trapezium, saw-tooth etc). Its Fourier components are evaluated and next, at discontinuity points slope and $y$ values are averaged out.
As to your last question. A circle of a circular toroid has finite curvature at point of vertical tangency. A cusp has infinite curvature at vertical tangent, but not a sub-category.
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
add a comment |
The first "curve" has DE $ dy/dx= cos y/ cos x,quad y = tan^{-1} (tan x ) $ with slope $+1$ everywhere. A regular continuous curve.
In second curve with a corner it has first degree contact i.e., same $(x,y)$, first and second degree values (slope,curvature) can be different. Here wave equation $ y= cos^{-1} (cos x), $ there is change in slope at the corner point $x=pi$. Slope goes from $1$ to $-1$ via $0$ with tangent parallel to x-axis.
Second example below of curve with a corner DE $ y^{''2} +1- y^2 =0 $ is a periodic curve. BC $ (0,-frac12) $. At the corner $(x approx 1.81,y=1), y^{'}= pm 1.59, y^{''}=0 $.
A cusp has first degree contact, same slope of infinite curvature whose sign changes at cusp location and the slope also passes through either zero or infinity value. Examples are point of contact of a circle when a cycloid type curve is produced by rolling circles on a straight line or another circle. Typically, last point of tracks when direction changes in an automobile in reverse gear.
There are curves that are synthesized from individual curves like discontinuous slope electrical wave-forms (square, trapezium, saw-tooth etc). Its Fourier components are evaluated and next, at discontinuity points slope and $y$ values are averaged out.
As to your last question. A circle of a circular toroid has finite curvature at point of vertical tangency. A cusp has infinite curvature at vertical tangent, but not a sub-category.
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
add a comment |
The first "curve" has DE $ dy/dx= cos y/ cos x,quad y = tan^{-1} (tan x ) $ with slope $+1$ everywhere. A regular continuous curve.
In second curve with a corner it has first degree contact i.e., same $(x,y)$, first and second degree values (slope,curvature) can be different. Here wave equation $ y= cos^{-1} (cos x), $ there is change in slope at the corner point $x=pi$. Slope goes from $1$ to $-1$ via $0$ with tangent parallel to x-axis.
Second example below of curve with a corner DE $ y^{''2} +1- y^2 =0 $ is a periodic curve. BC $ (0,-frac12) $. At the corner $(x approx 1.81,y=1), y^{'}= pm 1.59, y^{''}=0 $.
A cusp has first degree contact, same slope of infinite curvature whose sign changes at cusp location and the slope also passes through either zero or infinity value. Examples are point of contact of a circle when a cycloid type curve is produced by rolling circles on a straight line or another circle. Typically, last point of tracks when direction changes in an automobile in reverse gear.
There are curves that are synthesized from individual curves like discontinuous slope electrical wave-forms (square, trapezium, saw-tooth etc). Its Fourier components are evaluated and next, at discontinuity points slope and $y$ values are averaged out.
As to your last question. A circle of a circular toroid has finite curvature at point of vertical tangency. A cusp has infinite curvature at vertical tangent, but not a sub-category.
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
The first "curve" has DE $ dy/dx= cos y/ cos x,quad y = tan^{-1} (tan x ) $ with slope $+1$ everywhere. A regular continuous curve.
In second curve with a corner it has first degree contact i.e., same $(x,y)$, first and second degree values (slope,curvature) can be different. Here wave equation $ y= cos^{-1} (cos x), $ there is change in slope at the corner point $x=pi$. Slope goes from $1$ to $-1$ via $0$ with tangent parallel to x-axis.
Second example below of curve with a corner DE $ y^{''2} +1- y^2 =0 $ is a periodic curve. BC $ (0,-frac12) $. At the corner $(x approx 1.81,y=1), y^{'}= pm 1.59, y^{''}=0 $.
A cusp has first degree contact, same slope of infinite curvature whose sign changes at cusp location and the slope also passes through either zero or infinity value. Examples are point of contact of a circle when a cycloid type curve is produced by rolling circles on a straight line or another circle. Typically, last point of tracks when direction changes in an automobile in reverse gear.
There are curves that are synthesized from individual curves like discontinuous slope electrical wave-forms (square, trapezium, saw-tooth etc). Its Fourier components are evaluated and next, at discontinuity points slope and $y$ values are averaged out.
As to your last question. A circle of a circular toroid has finite curvature at point of vertical tangency. A cusp has infinite curvature at vertical tangent, but not a sub-category.
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
edited Oct 8 '17 at 6:22
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
answered Oct 7 '17 at 18:45
Narasimham
20.6k52158
20.6k52158
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%2f2429089%2fcusp-vs-corner-or-both%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
1
I think what they meant was that you have a cusp if the two parts are "tangent" to one another. In other words, if the angle they make is $0^circ$. In this case, however, the two curves clearly make a $45^circ$ angle, so I would think it should be called a corner. Still, the definitions aren't entirely clear on what to do in this case.
– Arthur
Sep 14 '17 at 10:31
Thanks! That's what I was thinking. By the definition of the corner, when we have a "regular" cusp, since the one-sided derivatives are different (one is -infinity and another being +infinity), would it be also called a corner?
– Harry Hong
Sep 14 '17 at 10:39
Is it different when it's really undefined? I dunno. To me the definitions are a bit unclear, so I don't know.
– Arthur
Sep 14 '17 at 10:41