Tag Archives: conic section

Halving a triangle, follow-up number one, ellipse

The previous post is “Analytic (coordinate) geometry has its good points, but elegance is not one of them”, in which a formula for any line cutting a triangle in half was found. The envelope of the cutting line for one section (of three) of the triangle was found to be always a hyperbola, which got me thinking “How do I get an ellipse?”. Clearly not by cutting a triangle in half, which involved taking two points A’ and B’on adjacent sides of the triangle, and making the product of their distances from the point of intersection of the sides equal to half the product of the lengths of the two sides.

So we cannot take two distances along two lines from the same point, lets try two distances from separate points on two lines, and keep the product of the distances constant. Magic:
envelope ellipse cropped
The base line is AF. Line DL is set parallel to AF. B and G are the two points of interest, where AB and DG are the two distances, and the envelope of the line BG is an ellipse which touches AF and DL. The really interesting thing about this is that the lines do not have to be parallel, and that as the points A and F are placed nearer and nearer to the point of intersection the ellipse becomes more and more hyperbolic at the nearby end.

The next post will be a different follow-up to the triangle halving.

Constructions made with GEOSTRUCT, an online browser application:

To get geostruct from the net click
http://www.mathcomesalive.com/geostruct/geostructforbrowser1.html

and to download the .doc instructions file
http://www.mathcomesalive.com/geostruct/geostruct basics.doc

 

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Filed under conic sections, construction, geometrical

Real problems with conic sections (ellipse, parabola) part two

So suppose we have a parabolic curve and we want to find out stuff about it.

Its equation … Oh, we have no axes.

Its focus … That would be nice, but it is a bit out of reach.

Its axis, in fact its axis of symmetry … Fold it in half? But how?

Try the method of part one, with the ellipse. (previous post)

parabola find the focus1

This looks promising. I even get another axis, for my coordinate system, if I really want the equation.

Now, analysis of the standard equation for a parabola (see later) says that a line at 45 deg to the axis, as shown, cuts the parabola at a point four focal lengths from the axis. In the picture, marked on the “vertical”axis, this is the length DH

parabola find the focus2

So I need a point one quarter of the way from D to H. Easy !

parabola find the focus3

and then the circle center D, with radius DH/4 cuts the axis of the parabola at the focal point (the focus).

Even better, we get the directrix as well …

parabola find the focus4

and now for the mathy bit (well, you do the algebra, I did the picture)

parabola find the focus math

Yes, I know that this one points up and the previous one pointed to the right !

All diagrams were created with my geometrical construction program, GEOSTRUCT

You will find it here:

www.mathcomesalive.com/geostruct/geostructforbrowser1.html

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Filed under conic sections, construction, engineering, geometry, teaching

GEOSTRUCT, a program for investigative geometry

I have been developing this computer software / program / application for some years now, and it is now accessible as a web page, to run in your browser.

It provides basic geometric construction facilities, with lines, points and circles, from which endless possibilities follow.

Just try it out, it’s free.

Click on this or copy and paste for later : www.mathcomesalive.com/geostruct/geostructforbrowser1.html

.Here are some of the basic features, and examples of more advanced constructions, almost all based on straightedge and compass, from “make line pass through a point” to “intersection of two circles”, and dynamic constructions with rolling and rotating circles.

help pic 1
Two lines, with points placed on them
help pic 3
Three random lines with two points of intersection generated
help pic 6
Five free points, three generated circles and a center point
help pic 7
Three free points, connected as point pairs, medians generated
help pic 5
Two free circles and three free points, point pairs and centers generated
gif line and circle
GIF showing points of intersection of a line with a circle
hypocycloid locus
Construction for locus of hypocycloid
circle in a segment
gif002
GIF showing a dilation (stretch) in the horizontal direction
gif piston cylinder
Piston and flywheel
gif touching2circles
Construction for circle touching two circles
gif parabola
Construction for the locus of a parabola, focus-directrix definition.

 

 

 

 

 

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Filed under education, geometry, math, operations, teaching

What they didn’t tell you at school (conic sections)

They tell you that conic sections are exactly what the words describe : Slice a double cone, the edge of the slice is a conic section, parabola, hyperbola, ellipse.

Then they tell you that y = x^2 is a parabola, or that all second degree equations in x and y are conic sections, or worst of all, they come up with the focus/directrix definition.

NOBODY shows you how to get the equation from the sliced cone !!!!!!!!!!!!

Well, here goes – (the math is after the picture, and it is so simple)

Image

Image

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Equations of conics

From the previous post (it is now the next one !) it can be seen that projection of the unit circle from the double cone vertex onto the slicing plane will give a second degree equation.  What a nice lead in to these equations, and also projective geometry, and to 3D xyz stuff.

The focus/directrix definition is so “rabbit out of a hat” that it’s time for its retirement. In any case, what use is the focus except for a parabola. And the directrix ? Surely “Directrix” is a female director !!!!!!!!!!!!

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Filed under algebra, geometry