Can parametric equations graph all kinds of lines?
I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example: 
and even
So naturally one would think that, if we put pen to paper, and drew something, anything, there would be a set of parametric equations to describe that shape or figure (of course, I'm not saying that finding the equations will be easy, but they should exist). So, is this true?
functions parametric
edited Apr 13 '17 at 12:21
Community♦
1
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
add a comment |
I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example:
and even
So naturally one would think that, if we put pen to paper, and drew something, anything, there would be a set of parametric equations to describe that shape or figure (of course, I'm not saying that finding the equations will be easy, but they should exist). So, is this true?
functions parametric
edited Apr 13 '17 at 12:21
Community♦
1
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
i.stack.imgur.com/EHNR8.png If you look at this image, an answer becomes a lot more apparent. You can define as many separate functions as you like over whatever intervals you like and adjoin them to each other by the "endpoints" of the intervals to get whatever shapes you like.
– ÍgjøgnumMeg
Mar 19 '16 at 5:03
@Ed_4434 yes, if you define them over different intervals, but is it possible to do something like that with a single set of equations?
– Airdish
Mar 19 '16 at 5:05
add a comment |
I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example:
and even
So naturally one would think that, if we put pen to paper, and drew something, anything, there would be a set of parametric equations to describe that shape or figure (of course, I'm not saying that finding the equations will be easy, but they should exist). So, is this true?
functions parametric
edited Apr 13 '17 at 12:21
Community♦
1
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example:
and even
So naturally one would think that, if we put pen to paper, and drew something, anything, there would be a set of parametric equations to describe that shape or figure (of course, I'm not saying that finding the equations will be easy, but they should exist). So, is this true?
functions parametric
functions parametric
edited Apr 13 '17 at 12:21
Community♦
1
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
edited Apr 13 '17 at 12:21
Community♦
1
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
edited Apr 13 '17 at 12:21
Community♦
1
1
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
asked Mar 19 '16 at 4:57
AirdishAirdish
1,261623
1,261623
i.stack.imgur.com/EHNR8.png If you look at this image, an answer becomes a lot more apparent. You can define as many separate functions as you like over whatever intervals you like and adjoin them to each other by the "endpoints" of the intervals to get whatever shapes you like.
– ÍgjøgnumMeg
Mar 19 '16 at 5:03
@Ed_4434 yes, if you define them over different intervals, but is it possible to do something like that with a single set of equations?
– Airdish
Mar 19 '16 at 5:05
add a comment |
i.stack.imgur.com/EHNR8.png If you look at this image, an answer becomes a lot more apparent. You can define as many separate functions as you like over whatever intervals you like and adjoin them to each other by the "endpoints" of the intervals to get whatever shapes you like.
– ÍgjøgnumMeg
Mar 19 '16 at 5:03
@Ed_4434 yes, if you define them over different intervals, but is it possible to do something like that with a single set of equations?
– Airdish
Mar 19 '16 at 5:05
i.stack.imgur.com/EHNR8.png If you look at this image, an answer becomes a lot more apparent. You can define as many separate functions as you like over whatever intervals you like and adjoin them to each other by the "endpoints" of the intervals to get whatever shapes you like.
– ÍgjøgnumMeg
Mar 19 '16 at 5:03
i.stack.imgur.com/EHNR8.png If you look at this image, an answer becomes a lot more apparent. You can define as many separate functions as you like over whatever intervals you like and adjoin them to each other by the "endpoints" of the intervals to get whatever shapes you like.
– ÍgjøgnumMeg
Mar 19 '16 at 5:03
@Ed_4434 yes, if you define them over different intervals, but is it possible to do something like that with a single set of equations?
– Airdish
Mar 19 '16 at 5:05
@Ed_4434 yes, if you define them over different intervals, but is it possible to do something like that with a single set of equations?
– Airdish
Mar 19 '16 at 5:05
add a comment |
1 Answer
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As far as I know, yes. Just about any line you could draw could be approximated by a parametric curve consisting of polynomials. For the Cartesian equivalent, look up "infinite series polynomials". A higher degree of accuracy can be achieved by adding more polynomial terms. Cusps (corners) could be achieved by including such functions as $sqrt{x^2}$. Jump discontinuities could be achieved by including such functions as $sqrt{x^2}/x$. A single parametric curve could then look as if it was multiple lines.
Disclaimer: All equations produced this way would be approximations of the originally drawn line, but could be made arbitrarily accurate. Coming up with these functions could be extremely tedious to do by hand, because any change made to any portion of the function would require adjustments to the rest of the curve.
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
add a comment |
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1 Answer
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As far as I know, yes. Just about any line you could draw could be approximated by a parametric curve consisting of polynomials. For the Cartesian equivalent, look up "infinite series polynomials". A higher degree of accuracy can be achieved by adding more polynomial terms. Cusps (corners) could be achieved by including such functions as $sqrt{x^2}$. Jump discontinuities could be achieved by including such functions as $sqrt{x^2}/x$. A single parametric curve could then look as if it was multiple lines.
Disclaimer: All equations produced this way would be approximations of the originally drawn line, but could be made arbitrarily accurate. Coming up with these functions could be extremely tedious to do by hand, because any change made to any portion of the function would require adjustments to the rest of the curve.
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
add a comment |
As far as I know, yes. Just about any line you could draw could be approximated by a parametric curve consisting of polynomials. For the Cartesian equivalent, look up "infinite series polynomials". A higher degree of accuracy can be achieved by adding more polynomial terms. Cusps (corners) could be achieved by including such functions as $sqrt{x^2}$. Jump discontinuities could be achieved by including such functions as $sqrt{x^2}/x$. A single parametric curve could then look as if it was multiple lines.
Disclaimer: All equations produced this way would be approximations of the originally drawn line, but could be made arbitrarily accurate. Coming up with these functions could be extremely tedious to do by hand, because any change made to any portion of the function would require adjustments to the rest of the curve.
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
add a comment |
As far as I know, yes. Just about any line you could draw could be approximated by a parametric curve consisting of polynomials. For the Cartesian equivalent, look up "infinite series polynomials". A higher degree of accuracy can be achieved by adding more polynomial terms. Cusps (corners) could be achieved by including such functions as $sqrt{x^2}$. Jump discontinuities could be achieved by including such functions as $sqrt{x^2}/x$. A single parametric curve could then look as if it was multiple lines.
Disclaimer: All equations produced this way would be approximations of the originally drawn line, but could be made arbitrarily accurate. Coming up with these functions could be extremely tedious to do by hand, because any change made to any portion of the function would require adjustments to the rest of the curve.
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
As far as I know, yes. Just about any line you could draw could be approximated by a parametric curve consisting of polynomials. For the Cartesian equivalent, look up "infinite series polynomials". A higher degree of accuracy can be achieved by adding more polynomial terms. Cusps (corners) could be achieved by including such functions as $sqrt{x^2}$. Jump discontinuities could be achieved by including such functions as $sqrt{x^2}/x$. A single parametric curve could then look as if it was multiple lines.
Disclaimer: All equations produced this way would be approximations of the originally drawn line, but could be made arbitrarily accurate. Coming up with these functions could be extremely tedious to do by hand, because any change made to any portion of the function would require adjustments to the rest of the curve.
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
edited Oct 21 '18 at 2:24
Xander Henderson
14.2k103554
14.2k103554
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
answered Oct 21 '18 at 1:41
Broadneck High SchoolBroadneck High School
1
1
add a comment |
add a comment |
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i.stack.imgur.com/EHNR8.png If you look at this image, an answer becomes a lot more apparent. You can define as many separate functions as you like over whatever intervals you like and adjoin them to each other by the "endpoints" of the intervals to get whatever shapes you like.
– ÍgjøgnumMeg
Mar 19 '16 at 5:03
@Ed_4434 yes, if you define them over different intervals, but is it possible to do something like that with a single set of equations?
– Airdish
Mar 19 '16 at 5:05