1. Keghvart Hagopian and Stephen Hemedes. California State University, Channel Islands
Research Mentor: Prof. James McDonough
1) Blitzer, Robert. "Point Slope Formula." Precalculus. 5th Ed. N.p.: Pearson, 2014. 190. Print.
2) “Plane Of Projection." Merriam-Webster.com. Merriam-Webster, 2016.
Web. 24 March 2016.
3) An Introduction to Abstract Mathematics / Robert Bond, William J. Keane. Long Grove, Ill.:
Waveland Press, 1999
4) http://study.com/academy/lesson/what-is-slope-definition-formulas-quiz.html
5) http://www.cut-the-knot.org/triangle/pythpar/Fifth.shtml
6) http://geomalgorithms.com/a05-_intersect-1.html
We are currently conducting a research study for our mathematical
logic and reason course with regards to the maximum and minimum
amounts of times lines can intersect without being parallel, or the
same line. We expand this concept to vertical planes and calculate
the maximum and minimum amounts of times vertical planes can
intersect at lines and points.
N = Number of Lines
Distinct Line – A line with different slopes and possibly different y-intercept
Distinct Crossing – A point where only two lines may cross
For N distinct lines with different slopes, the maximum number of distinct crossings is equal to
෍
𝑖=1
𝑁
(𝑖 − 1) =
𝑁 − 1 𝑁
2
For one line 𝐿1, there are no distinct crossings.
1 − 1 1
2
=
0 1
2
= 0
Proof:
Assume that for (N-1) distinct lines that are not parallel, that you have distinct crossings.
Choose another distinct line such that it crosses every other line at distinct crossings, this adds
(N-1) more distinct crossings.
෍
𝑖=1
𝑁−1
(𝑖 − 1) + 𝑁 − 1
This will equal…
෍
𝑖 = 1
𝑁
(𝑖 − 1) =
(𝑁 − 1)(𝑁)
2
Formula For Calculating Max # Intersections Of Lines
Formula For Calculating Min # Intersections Of Lines
Examples / Euclid's Fifth Postulate
N = Number of Lines
P = Minimum Number of Intersections
𝑁0 = 𝑃 𝑁 ≠ 1
We claim that;
If we have N distinct slopes 𝑚1, 𝑚2, 𝑚3 … 𝑚𝑁 we can choose a point 𝑥1 in which all lines intersect
at that point.
Proof
Let 𝑚𝑖 and 𝑚𝑗 be two distinct slopes ( i ≠ J)
Where 𝑖 ≤ 𝑁 and 𝑗 ≤ 𝑁
By the point slope formula, you will have two equations.
𝑦 – 𝑦1 = 𝑚𝑖(𝑥 − 𝑥1)
𝑦 – 𝑦1 = 𝑚𝑗(𝑥 − 𝑥1) 𝑦 – 𝑦1 = 𝑚𝑖(𝑥 – 𝑥1)
𝑦 – 𝑦1 = 𝑚 𝑛+1 𝑥 − 𝑚 𝑛+1 𝑥1 + 𝑦1 𝑦 – 𝑦1 = 𝑚1(𝑥1 – 𝑥1)
𝑦 – 𝑦1 = 𝑚𝑖 𝑥 – 𝑚𝑖 𝑥1 𝑦 = 𝑚 𝑛+1 𝑥 − 𝑚 𝑛+1 𝑥1 + 𝑦1 𝑦 – 𝑦1 = 0
𝑦 = 𝑚𝑖 𝑥 − 𝑚𝑖 𝑥1 + 𝑦1 𝑦 = 𝑦1
We set both equations equal to each other
𝑚 𝑛+1 𝑥 − 𝑚 𝑛+1 𝑥1 + 𝑦1 = 𝑚𝑖 𝑥 − 𝑚𝑖 𝑥1 + 𝑦1
𝑚 𝑛+1 𝑥 − 𝑚𝑖 𝑥 = 𝑚 𝑛+1 𝑥1 − 𝑚𝑖 𝑥1
𝑚 𝑛+1 − 𝑚𝑖 𝑥 = (𝑚 𝑛+1 − 𝑚𝑖)𝑥1
𝑥 = 𝑥1
Abstract
(3-1)(3) = 2 X 3= 3
2 2
30 = 1
Minimum # Crossings = 1
Maximum # Crossings = 3
(4-1)(4) = 3 X 4= 6
2 2
40 = 1
Minimum # Crossings = 1
Maximum # Crossings = 6
Vertical Planes
References
We can further generalize our proofs to vertical geometrical planes as planes are only lines that
exist in three dimensions.
y – y1 = mi(x−x1) + cZ
𝑦 – 𝑦1 = 𝑚𝑗(𝑥 − 𝑥1) + 𝑐𝑍
𝑦 = 𝑚𝑖 𝑥 + 𝑏 + 𝑐𝑍
The Z variable indicates three dimensional projection which can either be positive or negative.
Projection - a plane that is intersected by imaginary lines drawn from the eye to every point on the object and that
is therefore the plane on which the pictorial representation in perspective is formed
𝐿𝑒𝑡 𝑐 = 0,
𝑦 – 𝑦1 = 𝑚𝑖(𝑥 − 𝑥1) + 0 ⋅ 𝑍 𝑦 – 𝑦1 = 𝑚𝑖(𝑥 − 𝑥1)
𝑦 – 𝑦1 = 𝑚𝑗(𝑥 − 𝑥1) + 0 ⋅ 𝑍 𝑦 – 𝑦1 = 𝑚𝑗(𝑥 − 𝑥1)
𝑦 = 𝑚𝑖 𝑥 + 𝑏 + 0 ⋅ 𝑍 𝑦 = 𝑚𝑖 𝑥 + 𝑏
In 3D, a line L is either parallel to a plane P or intersects it in a single point.
∴ a projection of a vertical line in three dimension is a point in two dimensions.
Examples
Calculating Min / Max Intersections of Lines and Generalizing to Vertical Planes
Slope – is the steepness of a line.
Euclid's Fifth Postulate refers to the sum of two angles A and B formed by a line L and another two
lines 𝐿1 and 𝐿2 . If these interior angles sum up to less than 180°, then lines 𝐿1 and 𝐿2 must meet on
the side of angles A and B if continued indefinitely.
∴ if you have two lines that intersect at a point, you can use a transversal to pass through two
distinct points. Then if you take one of the lines and make it parallel to itself, the interior angle
formed (α and β) sum up to less than 180°, then the lines will eventually have another crossing.
180° - (α + β)
By 5)
By 3)
By 2,6
By 1