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Some mention should be made that a quadratic Bezier curve is a parabolic segment. - SharkD 20:06, 30 October 2006 (UTC)
From the equations:
and
and
surely it is obvious that f=r sin theta, using equations 1 and 3. Equation 2 is just confusing.
DOwenWilliams ( talk) 22:55, 15 July 2015 (UTC)
The assertion that all parabolas are similar to each other (or that they have the same shape) is incorrect, and I'm planning on fixing that soon. But I just wanted to say something here first.
In fact, the whole lead is of fairly poor quality and could use a complete rewrite. I'll possibly work on that, too, but more slowly. — Preceding unsigned comment added by 24.56.116.202 ( talk) 15:07, 16 August 2015 (UTC)
The pageview statistics for this article closely mirror activity in schools, with minima during school vacations and weekends. This presumably shows that many of the readers are school students. Editors should be aware of this. Don't make the article more sophisticated than its readers can appreciate. DOwenWilliams ( talk) 19:28, 16 August 2015 (UTC)
"Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids." This statement is obviously nonsense. Parabolic reflectors are, in fact, parabolic. What would they be called otherwise? "Paraboloidic"? This paragraph should be stricken from the article. — Preceding unsigned comment added by 50.255.2.98 ( talk) 00:17, 11 January 2017 (UTC)
It seems to me this is a lovely property of parabolas that has not previously been described in this entry on Parabolas. It has as much right to appear here as many of the other entries
Is there anybody else who can consider this entry or am is it to be damned on the whim of a single individual?
Obviously, I’m not familiar with all the subtleties but I do know that there are many contributions that relate to my field of expertise, Isaac Newton, that are very poor. Perhaps the people who have the power to remove items should have a look at those.
If there is anything wrong with this entry I will be happy to change it.
S is the Focus and V is the Principal Vertex of the parabola VG. Draw VX perpendicular to SV.
Take any point B on VG and drop a perpendicular BQ from B to VX. Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J.
For the parabola, the segment VBV, the area enclosed by the chord VB and the arc VB, is equal to ∆VBQ / 3, also
The Area of the Parabolic Sector SVB = ∆SVB + ∆VBQ / 3
Since triangles TSB and QBJ are similar:
Therefore, the Area of the Parabolic Sector , and can be found from the length of VJ, as found above.
It should be noted that a circle through S, V and B also passes through J.
Conversely, if a point, B on the parabola VG is to be found so that the Area of the Sector SVB is equal to a specified value, determine the point J on VX, and construct a circle through S, V and J. Since SJ is the diameter, the centre of the circle is at its mid-point, and it lies on the perpendicular bisector of SV, a distance of one half VJ from SV. The point required, B is where this circle intersects the parabola.
If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward. It follows that J moves at constant speed along VX as B moves along the parabola.
If the speed of the body at the vertex, where it is moving perpendicularly to SV is v, then the speed of J is equal to 3v/4.
The construction can be extended simply to include the case where neither radius coincides with the axis, SV as follows. Let A be a fixed point on VG between V and B, and point H be the intersection on VX with the perpendicular to SA at A. From the above, the Area of the Parabolic Sector
Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1 Proposition 16, Corollary 6 of the Principia, the speed of a body moving along a parabola with a force directed towards the focus is inversely as the square root of the radius. If the speed at A is v, then at the vertex, V it is , and point J moves at a constant speed of
Note: the above construction was devised by Isaac Newton and can be found in Book 1 of the Principia as Proposition 30.
— Preceding unsigned comment added by Mikerollem ( talk • contribs) 10:56, 14 September 2017 (UTC)
My physics master Dennis 'Dicky' Dyson in 1959 was a stickler for accuracy and for thinking things through from first principles irrespective of commonly held views. For example, he held scientists make new discoveries by 'Trial and success', not 'Trial and error'.
He taught me that a projectile moving freely under earth's gravity (neglecting air resistance) is in orbit round the earth - therefore its path (according to Newton) is that of a satellite, i.e. an ellipse with the centre of the earth at one focus. The lines of gravitational force are NOT parallel (which would produce a parabola) but converge towards the centre of the earth. That this effect is infinitesimal and unmeasurable over common distances, is immaterial - theoretically the path is a portion of an ellipse, even though it may very, very, very, closely approach a parabola; so closely you can't see the difference. But truth is truth!
86.187.168.50 ( talk) 23:34, 12 January 2018 (UTC)
At a size of over 74K this article has become too unwieldy and should be split. It has been suggested that the main readership of this page consists of students whose interests generally don't include some of the more esoteric topics that are included in the page. I would think that a natural split would break out those topics of interest to this population and have a second page devoted to more advanced (that is, requiring more background or being less directly relevant) topics. Any suggestions or other ideas? -- Bill Cherowitzo ( talk) 19:25, 24 January 2018 (UTC)
Note: This is about Special:Diff/905194553 ~ ToBeFree ( talk) 21:00, 8 July 2019 (UTC)
All the theory is fine, but for people looking for equations they can use in their daily lives it would be nice with a listing in the beginning for finding the vertex, foci, directrix and intersect. I realize that the addition I made was incomplete, but some kind of user guide before the reference guide would make the page useful for people like me. The reference that I added had exactly what I needed. It would be nice to have both in the wiki page.
By the way how do I comment on a change. I would rather have made this post as a comment.
Freeduck ( talk) 20:04, 8 July 2019 (UTC)
In the section "Axis of symmetry parallel to the y axis" to the right of the text "More generally, if the vertex is ..." There is a plot called, "Parabola: general case". Given the matching text and the fact they appear together on the same horizontal position it is natural to assume the paragraph goes with the plot -- but it does not. That text is actually describing the previous plot. This is confusing.
Lower down, even past the subsection, "remarks¨, there is another full section called "General Case¨. That is where the "Parabola: general" case plot belongs. I hope someone will move the plot down. 86.233.234.10 ( talk) 15:47, 2 July 2021 (UTC)
I'm not an expert in math, but isn't the sentence "The graph of a quadratic function y=ax²+bx+c is a parabola if x!=0 and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis." saying the same thing twice, or is there a particular usefulness to list the converse? -- laagone talk 13:37, 12 March 2024 (UTC)
This
level-4 vital article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
This article is substantially duplicated by a piece in an external publication. Please do not flag this article as a copyright violation of the following source:
|
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 10 sections are present. |
Some mention should be made that a quadratic Bezier curve is a parabolic segment. - SharkD 20:06, 30 October 2006 (UTC)
From the equations:
and
and
surely it is obvious that f=r sin theta, using equations 1 and 3. Equation 2 is just confusing.
DOwenWilliams ( talk) 22:55, 15 July 2015 (UTC)
The assertion that all parabolas are similar to each other (or that they have the same shape) is incorrect, and I'm planning on fixing that soon. But I just wanted to say something here first.
In fact, the whole lead is of fairly poor quality and could use a complete rewrite. I'll possibly work on that, too, but more slowly. — Preceding unsigned comment added by 24.56.116.202 ( talk) 15:07, 16 August 2015 (UTC)
The pageview statistics for this article closely mirror activity in schools, with minima during school vacations and weekends. This presumably shows that many of the readers are school students. Editors should be aware of this. Don't make the article more sophisticated than its readers can appreciate. DOwenWilliams ( talk) 19:28, 16 August 2015 (UTC)
"Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids." This statement is obviously nonsense. Parabolic reflectors are, in fact, parabolic. What would they be called otherwise? "Paraboloidic"? This paragraph should be stricken from the article. — Preceding unsigned comment added by 50.255.2.98 ( talk) 00:17, 11 January 2017 (UTC)
It seems to me this is a lovely property of parabolas that has not previously been described in this entry on Parabolas. It has as much right to appear here as many of the other entries
Is there anybody else who can consider this entry or am is it to be damned on the whim of a single individual?
Obviously, I’m not familiar with all the subtleties but I do know that there are many contributions that relate to my field of expertise, Isaac Newton, that are very poor. Perhaps the people who have the power to remove items should have a look at those.
If there is anything wrong with this entry I will be happy to change it.
S is the Focus and V is the Principal Vertex of the parabola VG. Draw VX perpendicular to SV.
Take any point B on VG and drop a perpendicular BQ from B to VX. Draw perpendicular ST intersecting BQ, extended if necessary, at T. At B draw the perpendicular BJ, intersecting VX at J.
For the parabola, the segment VBV, the area enclosed by the chord VB and the arc VB, is equal to ∆VBQ / 3, also
The Area of the Parabolic Sector SVB = ∆SVB + ∆VBQ / 3
Since triangles TSB and QBJ are similar:
Therefore, the Area of the Parabolic Sector , and can be found from the length of VJ, as found above.
It should be noted that a circle through S, V and B also passes through J.
Conversely, if a point, B on the parabola VG is to be found so that the Area of the Sector SVB is equal to a specified value, determine the point J on VX, and construct a circle through S, V and J. Since SJ is the diameter, the centre of the circle is at its mid-point, and it lies on the perpendicular bisector of SV, a distance of one half VJ from SV. The point required, B is where this circle intersects the parabola.
If a body traces the path of the parabola due to an inverse square force directed towards S, the area SVB increases at a constant rate as point B moves forward. It follows that J moves at constant speed along VX as B moves along the parabola.
If the speed of the body at the vertex, where it is moving perpendicularly to SV is v, then the speed of J is equal to 3v/4.
The construction can be extended simply to include the case where neither radius coincides with the axis, SV as follows. Let A be a fixed point on VG between V and B, and point H be the intersection on VX with the perpendicular to SA at A. From the above, the Area of the Parabolic Sector
Conversely, if it is required to find the point B for a particular area SAB, find point J from HJ and point B as before. By Book 1 Proposition 16, Corollary 6 of the Principia, the speed of a body moving along a parabola with a force directed towards the focus is inversely as the square root of the radius. If the speed at A is v, then at the vertex, V it is , and point J moves at a constant speed of
Note: the above construction was devised by Isaac Newton and can be found in Book 1 of the Principia as Proposition 30.
— Preceding unsigned comment added by Mikerollem ( talk • contribs) 10:56, 14 September 2017 (UTC)
My physics master Dennis 'Dicky' Dyson in 1959 was a stickler for accuracy and for thinking things through from first principles irrespective of commonly held views. For example, he held scientists make new discoveries by 'Trial and success', not 'Trial and error'.
He taught me that a projectile moving freely under earth's gravity (neglecting air resistance) is in orbit round the earth - therefore its path (according to Newton) is that of a satellite, i.e. an ellipse with the centre of the earth at one focus. The lines of gravitational force are NOT parallel (which would produce a parabola) but converge towards the centre of the earth. That this effect is infinitesimal and unmeasurable over common distances, is immaterial - theoretically the path is a portion of an ellipse, even though it may very, very, very, closely approach a parabola; so closely you can't see the difference. But truth is truth!
86.187.168.50 ( talk) 23:34, 12 January 2018 (UTC)
At a size of over 74K this article has become too unwieldy and should be split. It has been suggested that the main readership of this page consists of students whose interests generally don't include some of the more esoteric topics that are included in the page. I would think that a natural split would break out those topics of interest to this population and have a second page devoted to more advanced (that is, requiring more background or being less directly relevant) topics. Any suggestions or other ideas? -- Bill Cherowitzo ( talk) 19:25, 24 January 2018 (UTC)
Note: This is about Special:Diff/905194553 ~ ToBeFree ( talk) 21:00, 8 July 2019 (UTC)
All the theory is fine, but for people looking for equations they can use in their daily lives it would be nice with a listing in the beginning for finding the vertex, foci, directrix and intersect. I realize that the addition I made was incomplete, but some kind of user guide before the reference guide would make the page useful for people like me. The reference that I added had exactly what I needed. It would be nice to have both in the wiki page.
By the way how do I comment on a change. I would rather have made this post as a comment.
Freeduck ( talk) 20:04, 8 July 2019 (UTC)
In the section "Axis of symmetry parallel to the y axis" to the right of the text "More generally, if the vertex is ..." There is a plot called, "Parabola: general case". Given the matching text and the fact they appear together on the same horizontal position it is natural to assume the paragraph goes with the plot -- but it does not. That text is actually describing the previous plot. This is confusing.
Lower down, even past the subsection, "remarks¨, there is another full section called "General Case¨. That is where the "Parabola: general" case plot belongs. I hope someone will move the plot down. 86.233.234.10 ( talk) 15:47, 2 July 2021 (UTC)
I'm not an expert in math, but isn't the sentence "The graph of a quadratic function y=ax²+bx+c is a parabola if x!=0 and, conversely, a parabola is the graph of a quadratic function if its axis is parallel to the y-axis." saying the same thing twice, or is there a particular usefulness to list the converse? -- laagone talk 13:37, 12 March 2024 (UTC)