20220902, 17:50  #1 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7^{2}·47 Posts 
Parallel Parabolas
It occurred to me, I thought about it, I googled it and found a lot of nonsensical "opinions"/definitions.
I think that any parallel parabola to any given parabola will have to superimpose it and hence be the same (Unless it is on a separate 2D Plane). I think the only valid definition of parallel curves is that any straight line perpendicular to one will be perpendicular to any other. Furthermore I think there can not exist any separate curve on a single plane (parabolic or otherwise) that is parallel to any given parabola. What do you think? https://www.jstor.org/stable/3027202 https://www.reddit.com/r/math/commen...to_each_other/ 
20220902, 18:16  #2 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3^{3}·263 Posts 
To generalize parallel to curves, consider concentric circles of differing radii in the same plane.
The shortest distance between any selected point on one circle and the nearest point on the other is a segment of a radial line. https://www.dictionary.com/browse/parallel One of the responses here indicates a curve parallel in the same plane to a parabola will be a sixth order polynomial. Last fiddled with by kriesel on 20220902 at 18:16 
20220902, 19:36  #3  
Feb 2017
Nowhere
3^{2}×677 Posts 
See also ENVELOPE OF A FAMILY OF PLANE CURVES
Quote:


20220902, 22:36  #4  
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2^{4}·5·83 Posts 
Quote:
Bézier curves have been studied extensively. A degree two Bézier curve is a parabolic section, and constant distance offsetting from a parabola is known to need degree six polynomials. Last fiddled with by retina on 20220902 at 22:41 

20220903, 00:41  #5 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7^{2}·47 Posts 
I am sorry but I take an issue with using OffsetLines to be equivalent/interchangeable with ParallelLines, despite the Wikipedia article stating just that:
https://en.wikipedia.org/wiki/Parallel_curve The examples there contradict each other. Please see the attached screenshot. In the image EQD170A which I made using AutoCAD's "Offset" command only the outer 4 polylines are parallel to the outermost polyline. not the inner 3 rectangles. The issue with them is that if you offset them outwards after they loose their round corners they create rectangles which are different than the original filleted polylines. so you can have 2 different offsets of the same "curve" one with the new sharp corners and the original rounded corners at the same distance (and on the same side). I don' think you can have 2 different parallel "curves" to a single curve at equal distances and on the same side. If you could then you could show that all closed/open curves on a 2D plane are parallel to all other closed/open curves on that plane. And that would be nonsense. Last fiddled with by a1call on 20220903 at 00:43 
20220903, 16:23  #6 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
3^{3}×263 Posts 
parabola 1: z=z_{1}
parabola 2: z_{2} != z_{1} https://en.wikipedia.org/wiki/Parabo...er_Quadric.png Last fiddled with by kriesel on 20220903 at 16:28 
20220903, 17:17  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
7^{2}×47 Posts 
Yes those are all identical parabolas on parallel 2D planes. I also considered parallel cross sections of cones forming parabolas, see 4th image below:
https://www.mechamath.com/geometry/a...onsofacone/ Which raises the question: If we define a perpendicular line to a curve at a given point by a perpendicular line to the tangent line to that curve at that point, how would you define a tangent line to a parabola on a plane other than the plane the parabola is drawn on? ETA: I assume you could still define a perpendicular line as a perpendicular to the tangent on the original plane. Re: Parallel == Offset, Consider any (non square) rectangle and square on a plane whose all edges are parallel and do not cross. They Can not be obtained in a single Offset from one another. Are they not parallel? Last fiddled with by a1call on 20220903 at 17:34 
20220905, 02:31  #8 
"Rashid Naimi"
Oct 2015
Remote to Here/There
100011111111_{2} Posts 
FWIW:
So in conclusion: * If the offset is not large enough and internal to eliminate the vertex curve there can parallel curves to a parabola * Parabolas formed by parallel slicing planes to a cone do not create parallel Parabolas Last fiddled with by a1call on 20220905 at 02:40 
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