Difference between revisions of "Formulas relating the number of lines, sections, and intersection points"

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\text{Some formulas relating the number of intersections, lines, and sections in a plane.}
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Some formulas relating the number of intersections, lines, and sections in a plane.
\text{1) The maximum number of intersection points of n lines is <math>\frac{n(n-1)}{2}</math>.}
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1) The maximum number of intersection points of n lines is <math>\frac{n(n-1)}{2}</math>.
 
Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2
 
Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2
 
new intersection points, and so on.
 
new intersection points, and so on.
\text{2) The maximum number of planar sections (assuming the plane is bound at some point) created by n lines is \frac{n^2+n+2}{2}}
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2) The maximum number of planar sections (assuming the plane is bound at some point) created by n lines is <math>\frac{n^2+n+2}{2}</math>
 
Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through  
 
Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through  
 
two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the  
 
two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the  
 
plane. (someone turn this into a formal proof with induction please)
 
plane. (someone turn this into a formal proof with induction please)
\text{3)The number of lines + number of intersection points = number of sections -1 assuming that no three lines intersect at one point.}
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3)The number of lines + number of intersection points = number of sections -1 assuming that no three lines intersect at one point.
 
The proof can be made by bashing the equation with the previous formulas
 
The proof can be made by bashing the equation with the previous formulas
\text{4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections created, provided that no three curved or straight lines intersect at the same point.}
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4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections created, provided that no three curved or straight lines intersect at the same point.
 
Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line.
 
Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line.

Revision as of 17:18, 4 June 2013

Some formulas relating the number of intersections, lines, and sections in a plane. 1) The maximum number of intersection points of n lines is $\frac{n(n-1)}{2}$. Proof: Each line intersects all the other lines to create the maximum number of intersections. The first line intersects n-1 lines, the second one has n-2 new intersection points, and so on. 2) The maximum number of planar sections (assuming the plane is bound at some point) created by n lines is $\frac{n^2+n+2}{2}$ Proof: Before there are any lines, there is one section. The first line passes through one section, splitting it in two. The second line passes through two sections, splitting it into a total of four sections. The third line passes through a maximum of three existing sections, adding three new sections to the plane. (someone turn this into a formal proof with induction please) 3)The number of lines + number of intersection points = number of sections -1 assuming that no three lines intersect at one point. The proof can be made by bashing the equation with the previous formulas 4) The number of curved lines (enclosed circles)+ number of lines+ number of intersection points -1= number of sections created, provided that no three curved or straight lines intersect at the same point. Proof: Almost the same, note however that two curved lines and intersect at two points, same for one curved and one straight line.