\begin{align} We also assume that a line is given by a point and a (nonzero) direction vector (in either direction). How can I make the seasons change faster in order to shorten the length of a calendar year on it? rev 2020.11.24.38066, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Thanks, Kumar. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. is perpendicular to the first line; therefore, $\color{#E06800}{(-40,-40,200)}\cdot u$ is constant along the first line. In this case, the constant for each line is $0$, so the lines intersect. If this constant is not the same for each line, the lines do not intersect. $$ But I'm getting a problem for $(6,8,4) (12,15,4)$ and $(6,8,2) (12,15,6)$. Kumar: I included in my answer below a treatment of the "zero coordinate of direction numbers" case you refer to in your EDIT. To compute the point of intersection, find the $t$ so that Do ETFs move on their own? \color{#C00000}{((10,10,6)-(5,5,4))}\times\color{#00A000}{((10,10,3)-(5,5,5))} is perpendicular to both lines; therefore, $\color{#0000FF}{(28,-24,0)}\cdot u$ is constant along each line. Line 2: points $(6,8,2),\ (12,15,6);$ directions $(6,7,4)$, equation is perpendicular to both lines; therefore, $\color{#0000FF}{(-20,20,0)}\cdot u$ is constant along each line. I need to find the intersection point (if any) of two line segments not simple lines. ). What do you mean by not getting correct values? \color{#C00000}{(5,5,2)}\times\color{#0000FF}{(-20,20,0)}=\color{#E06800}{(-40,-40,200)} Therefore ${\bf g} = \vec{CD} = (0,0,-2)$. $$\frac{x-5}{5}=\frac{y-5}{5}=\frac{z-5}{-2}.$$ \color{#00A000}{(6,8,2)}+\color{#00A000}{(6,7,4)}\cdot1/2=\left(9,\frac{23}{2},4\right) If you want to check, we can, in the second example, take $C = (6,8,4)$, $D = (6,8,2)$ “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2/4/9 UTC (8:30PM…, how to find specific point on edge of a triangle, Finding two points that have a defined distance between two intersecting lines, Find out intersection of lines with only edge points, Find a equidistant point between two intersecting lines. $\color{#C00}{0}$: since $(-20,20,0)\cdot u$ is constant along each of the first two lines, we just plug in a point from each line: $(-20,20,0)\cdot(5,5,4)=0$ and $(-20,20,0)\cdot(5,5,5)=0$. There's a problem above in Line 28 - what's the correct line? which point in the same direction; so we take the $+$ sign. Let ${\bf g} = \vec{CD}$. The same goes for $\color{#C00}{-24}$ and $\color{#C00}{-1360}$ from the second example. Someone can share some knowledge on a good algorithm that helps you know if two segments intersect? A Bitcoin locking script to force a certain payment to the receiver? A Unity ID allows you to buy and/or subscribe to Unity products and services, shop in the Asset Store and participate And using the given point $(5,5,5)$ of the second line gives its symmetric form $$ If you plug this into the second line equation you will be well on your way. $$\frac{x-5}{5}=\frac{y-5}{5}=\frac{z-4}{2}.$$ -680-1360t=-1360\\ If ${\bf f} \times {\bf g}$ and ${\bf f} \times {\bf e}$ point in the same direction, then the sign is $+$; Finding Points of Intersection of Two Lines. The first line has direction numbers $(5,5,2)$ while the second line has direction numbers $(5,5,-2).$ Once one has direction numbers $(a,b,c)$, one can use either given point of the line to obtain the symmetric form of its equation as $$\frac{z-4}{2}=\frac{z-5}{-2}$$ When i tested this with some other points i am not getting correct values. f*0.5f : -f*0.5f; // intersection.y = p1.y + (num+offset) / f; /// Calculate the intersection point of two line segments, /// Epsilon in pixels, //Debug.Log (string.Format("LineSegmentsIntersection2 p1 {0} p2 {1} p3 {2} p4{3}", p1, p2, p3, p4)); // the float value precision in the log is just 0.0f, http://books.google.com/books?id=xm...Faster Line Segment Intersection&pg=RA1-PA199, http://wiki.unity3d.com/index.php/3d_Math_functions, https://www.habrador.com/tutorials/math/5-line-line-intersection/, (You must log in or sign up to reply here. &=\color{#C00000}{(6,7,0)}\times\color{#00A000}{(6,7,4)}\\ ). &=\color{#0000FF}{(28,-24,0)} $$ $$\frac{x-6}{6}=\frac{y-8}{7}=\frac{z-4}{0},$$ Please expline is there any other conserns are there to solve this problem, This will helps a lot for me. In the case of two non-parallel lines, the intersection will always be on the lines somewhere. If this constant is not the same for each line, the lines do not intersect. What modern innovations have been/are being made for the piano, Baby proofing the space between fridge and wall. and ${\bf e} = (6,7,0)$ and ${\bf f} = (6,7,4)$. Problem Solving >. What if the lines don't perfectly intersect, will this method still give the "closest" point of intersection between the two lines? Method 1: Substitution. Category with zero morphisms implies zero object? The general point along the second line is Choosing US House delegation by winner-take-all statewide vote? Subtracting these we get, (a 1 b 2 – a 2 b 1) x = c 1 b 2 – c 2 b 1. I have two lines $(5,5,4) (10,10,6)$ and $(5,5,5) (10,10,3)$ with same $x$, $y$ and difference in $z$ values. How would sailing be affected if seas had actually dangerous large animals. \color{#E06800}{(0,0,-340)}\cdot(\color{#00A000}{(6,8,2)}+\color{#00A000}{(6,7,4)}t)=-1360\\ &=\color{#C00000}{(5,5,2)}\times\color{#00A000}{(5,5,-2)}\\ In a way the occurrence of a zero makes things easier since one knows immediately one of the variable values. The direction numbers $(a,b,c)$ for a line in space may be obtained from two points on the line by subtracting corresponding coordinates. But the more ways you can solve a problem, the better you probably understand the underlying mathematics. So I'll share a vector method. Method 2: Addition/Subtraction. We can arrive at the solution without the use of coordinates, except as a matter of calculation of the final answer. How can I find the location on the Z axis where two skew lines pass closest to each other on the XY plane? Example Questions. What is the consistency strength of this large cardinal? But in the case of line segments or rays which have a limited length, they might not actually intersect. Thanks for the reply. Line 1: points $(6,8,4),\ (12,15,4);$ directions $(6,7,0)$, "equation" Note: This gives the point of intersection of two lines, but if we are given line segments instead of lines, we have to also recheck that the point so computed actually lies on both the line segments. This gives us the value of x. \begin{align} For your second problem, the first line has a constant z value of 4. otherwise, the sign is $-$. there is also a whole mass of stuff regarding intersecting points/lines/plane/whatever on, // offset = same_sign(num,f) ? The same goes for $\color{#C00}{-24}$ and $\color{#C00}{-1360}$ from the second example. In this case, that constant is $-1360$. f*0.5f : -f*0.5f; // round direction //. \color{#00A000}{(5,5,5)}+\color{#00A000}{(5,5,-2)}t Since these two are the same, the lines intersect. $$, The second example $$ Some problems are easier one way, and this one is definitely easier via coordinates. At the point where you find $C+(1/2)e$ the calculation for second coord. Let $C$, $D$ be points on $\alpha$, $\beta$, resp. \color{#00A000}{(6,8,2)}+\color{#00A000}{(6,7,4)}t The intersection point of two lines is determined by segments to be calculated in one line: Vector_2D R = (r0 * (R11^R10) - r1 *(R01^R00)) / (r1^r0); And once the intersection point of two lines has been determined by the segments received, it is easy to estimate if the point belongs to the segments with the scalar product calculation as in the Background part of this article prescribed. in the Unity community. ADDED CASE: $\color{#C00}{400}$: since $(-40,-40,200)\cdot u$ is constant along the first line, we again just plug in a point from that line: $(-40,-40,200)\cdot(5,5,4)=400$. This would come out as some contradictory values in the above mechanical procedure. Let ${\bf e}$ and ${\bf f}$ be direction vectors of $\alpha$, $\beta$, resp. (or depending on the notation you prefer, $\vec{OM} = \vec{OC} \pm {\bf m}$), Did Star Trek ever tackle slavery as a theme in one of its episodes? There are at least two ways to approach such problems, through vectors as such and through coordinates.