Smoothly merging two parametric curves
Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I want to fit this $c(s)$ curve (say, a quadratic polynomial), so that it smoothly merges with $t(s)$. Basically, as $s$ increases, $c(s)$ and $t(s)$ should end up being pretty much the same. For simplicity, let's say that both $t(s)$ and $c(s)$ are quadratic curves.
I am guessing I need to optimise some kind of objective involving both the distance between $c(s)$ and $t(s)$, as well as the derivative of $c(s)$. Of course, there will be a hyperparameter to control how smooth we want to be, vs. how fast we want the merging to happen.
The figure below shows what I mean by "smooth merging".
Is there any established method to deal with this problem?
I also need this to be computationally fairly fast, but that shouldn't be a problem for a 1D problem.
regression parametric curves parameter-estimation
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
add a comment |
Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I want to fit this $c(s)$ curve (say, a quadratic polynomial), so that it smoothly merges with $t(s)$. Basically, as $s$ increases, $c(s)$ and $t(s)$ should end up being pretty much the same. For simplicity, let's say that both $t(s)$ and $c(s)$ are quadratic curves.
I am guessing I need to optimise some kind of objective involving both the distance between $c(s)$ and $t(s)$, as well as the derivative of $c(s)$. Of course, there will be a hyperparameter to control how smooth we want to be, vs. how fast we want the merging to happen.
The figure below shows what I mean by "smooth merging".
Is there any established method to deal with this problem?
I also need this to be computationally fairly fast, but that shouldn't be a problem for a 1D problem.
regression parametric curves parameter-estimation
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
add a comment |
Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I want to fit this $c(s)$ curve (say, a quadratic polynomial), so that it smoothly merges with $t(s)$. Basically, as $s$ increases, $c(s)$ and $t(s)$ should end up being pretty much the same. For simplicity, let's say that both $t(s)$ and $c(s)$ are quadratic curves.
I am guessing I need to optimise some kind of objective involving both the distance between $c(s)$ and $t(s)$, as well as the derivative of $c(s)$. Of course, there will be a hyperparameter to control how smooth we want to be, vs. how fast we want the merging to happen.
The figure below shows what I mean by "smooth merging".
Is there any established method to deal with this problem?
I also need this to be computationally fairly fast, but that shouldn't be a problem for a 1D problem.
regression parametric curves parameter-estimation
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
Let's imagine that an object follows a path described by the known parametric curve $t(s)$ for $s geq 0$. Now, another object follows another curve $c(s)$, that goes through a known point $c_0$. I want to fit this $c(s)$ curve (say, a quadratic polynomial), so that it smoothly merges with $t(s)$. Basically, as $s$ increases, $c(s)$ and $t(s)$ should end up being pretty much the same. For simplicity, let's say that both $t(s)$ and $c(s)$ are quadratic curves.
I am guessing I need to optimise some kind of objective involving both the distance between $c(s)$ and $t(s)$, as well as the derivative of $c(s)$. Of course, there will be a hyperparameter to control how smooth we want to be, vs. how fast we want the merging to happen.
The figure below shows what I mean by "smooth merging".
Is there any established method to deal with this problem?
I also need this to be computationally fairly fast, but that shouldn't be a problem for a 1D problem.
regression parametric curves parameter-estimation
regression parametric curves parameter-estimation
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
asked Nov 27 '18 at 10:25
francoisrfrancoisr
1062
1062
add a comment |
add a comment |
0
active
oldest
votes
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%2f3015607%2fsmoothly-merging-two-parametric-curves%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3015607%2fsmoothly-merging-two-parametric-curves%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
