Examples of triangulation

What Is Triangulation Psychology? | BetterHelp

While we'd like to believe that most everyone strives to treat others as they themselves would like to be treated, there are many people out there who seek to manipulate others for their personal gain. One such manipulation tactic is called triangulation. Fortunately, there are ways to spot the signs of triangulation so that you can avoid it. In this article, we'll explain triangulation, offer some situational examples, and show how you can get past it.

Triangulation In Psychology Refers To A Manipulation Tactic

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What Is Triangulation?

Triangulation is a manipulation tactic used to avoid a direct conversation. As the name suggests, it involves three parties.

Let's say Jake wants to communicate with Mary. Instead of directly talking to her, Jake will instead use Mary's brother, Jeff, as a relay. This allows Jake to control the communication they have. Triangulation is a tactic people use for many reasons. It can be used to create a rivalry or to make people turn against each other.

Odds are, you have probably experienced triangulation before. This way, they can separate the two of you and only involve you in the conversation as they see fit.

Triangulation As A Manipulation Tactic

Someone who is narcissistic or manipulative may use triangulation to control the narrative. For example, let's say you know someone who always lies. How can someone who is consistently dishonest gain some credibility? Say those friends believe the lies. You ask these friends about it and they get defensive.

This can create hostility or a general sense of unrest. No one likes to be questioned and no one wants to be involved in the squabble between you and another person. This has created a triangle that is difficult to break. Now your friends go back to the dishonest individual and start telling them you’re asking questions and don’t trust that person. Suddenly, you're the villain.

That's how triangulation works. If you have a grievance against someone, they will use other people to discourage opposition. If you try to, you're the bad guy. You may even start to believe they're telling the truth due to peer pressure and no one else seeing or realizing the issue. You may wonder, “Am I the only one seeing this? Is it real or just in my head, then?”

How To Avoid A Situation Like This

It can be difficult to avoid triangulation, but try to involve as few people as possible. When you confront someone, make sure no people are around or between you and the other person. If the individual tells you that other people believe them, don't try to talk to those people.

Sometimes, it's just not worth it to expose a toxic person. If they are a chronic liar, they will eventually lose in the end. It also depends on your standing. In a friend's circle or the family circle, if you are the black sheep of the group, trying to point out the lies of another person may make you feel more alienated.

As for the toxic individual in question, take a look at them. Do they have mental health issues, or are they victims of themselves or a past trauma? If so, it may not be worth confronting them. Instead, they should get the help they need.

In cases like these, triangulation is an intentional act. Someone knows how to play people and pit them against each other. They realize that if it's one versus one, they may lose. However, if they can recruit others to dogpile you, then you may feel like there's no way you can win. That's the entire point of triangulation. With that said, triangulation can be an unintentional act as well. Let's look at a few examples of that.

An Unintentional Tactic?

The problem with manipulation tactics is that sometimes they are completely intentional and other times they are not. People may do it without realizing it. This makes it hard for them to understand their actions and their negative effects on others.

People who do this tend to be awkward, shy, and afraid of confrontation. They may have no insight as to what they are doing. You probably know someone like this, or maybe you've done it yourself. As a child, you may have been wronged. Instead of talking to the person who wronged you, you may have told your parents, a teacher, or someone else. They may have talked to that person as a result and thus, the triangulation has begun.

It's an unconscious action and the reason we do it is that we like to be in tribes. We want to ally with someone else who agrees someone wronged you because you will then have more people on your side who can support you. It feels safe. Sometimes, in arguments, numbers beat an actual argument.

Manipulating someone via this method may feel satisfying in the moment but once the issue is resolved, it can be difficult to break the triangle. Say you and the other person resolve the issue, forgive each other, and move on. How about the other person you told? Will they do the same? If not, there is still fallout from the grudge being held by the third party. If the two of you had settled it one on one, it would be much less messy.

Three Points Of Triangulation

Triangulation can be separated into three parts. Those parts are:

The victim. This is the person who has a victim complex and will try to give the "woe is me" vibe. The person may be a victim or they may not, but either way, they are the person who makes the triangle. They tell others they are a victim, generate a sense of empathy from others, and thus generate the triangle.
The manipulator. This is the person who goes after those in the triangle.
The third participant. This is someone who is either in or out the triangle. They are the ones who are the savior of the victim, if you will.

A person who intentionally uses triangulation will often reenact the triangle whenever they can so as to build as many “support” networks for themselves as they can. They will create triangles at work, in school, within their family, and any other circle they have. They may do this for a variety of reasons—perhaps they love the chaos, or they enjoy being the victim and have a complex based around that, or they genuinely feel that using triangulation is a positive communication strategy.

Triangulation And Therapy

Triangulation is a natural part of therapy. You talk to your therapist about people who have wronged you. When it comes to couples therapy, triangulation is really the only way to go about things. The therapist is the one who has to intervene between two people who are arguing, not connecting, miscommunicating, etc. Often, the therapist can become the victim of the redirected anger between the two. In truth, the therapist does not take a side, and is instead supposed to be the mediator. However, they can find it hard to be in the middle of two people who each have their own side of the story. They should, nonetheless, strive to remain objective.

Triangulation In Psychology Refers To A Manipulation Tactic

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The goal is to be objective. Someone should not believe someone else just because they favor one person or the other; instead they need to learn how to be objective, take a step back, and look at the facts. For therapists, or even people who are involved in the triangle, they need to learn how to be factual, or just not be involved at all. Remaining neutral to a situation can sometimes be a good thing. Don't feel obligated to get involved or take someone’s side just because you're friends with them. Sometimes, they can use that friendship to pull you into a triangle that can be hard to leave.

How To Stop Manipulation

Recognizing and stopping manipulation are two different things, but you will need to be able to do both in order to completely remove it from your life. Here are a few ways you can begin to stop manipulation.

Set Boundaries. You know what you are willing to accept into your life and what you do not want. A manipulator might be able to bypass some of the walls you've built, but once you are aware of their game, you can make sure to fortify those walls and redefine your concept of respect. When you set boundaries, it makes it a lot harder for others to manipulate you.
Say No And Stay Safe. It's rare that a manipulator is going to change their ways immediately, even if they reach out for help. If it is someone you must engage with daily, make sure to clearly set and keep your boundaries firm with yourself and with that person. They can only manipulate you if you let them in.
Seek Help. If you're having a hard time dealing with people and need to learn how to better stand up for yourself, there is zero shame in speaking to a counselor about any problems you have. A licensed counselor can help you with your problems and teach you how to confront others in healthy ways. By speaking to a therapist, you can get the strength to be a more communicative person who doesn't have to resort to triangles to solve problems.

Dealing With Conflict Through BetterHelp

Studies have shown that online therapy can be a useful component of a treatment plant for those experiencing emotional turmoil arising out of their social circle. In a study published in the Journal of Medical Internet Research, the effects of online cognitive-behavioral therapy (CBT) on symptoms of social anxiety disorder were evaluated. The results of the report—a five-year follow-up on an earlier study—showed that online CBT created sustained and long-lasting results, with participants experiencing decreased social phobia even five years after treatment. Cognitive-behavioral therapy helps individuals cope with anxiety by increasing their understanding of potential triggering thoughts and behaviors, so that certain interactions, such as a painful confrontation, can be more manageable.

As discussed above, if you’re being manipulated in a relationship, online therapy can help you. Using BetterHelp, you can connect with a certified therapist at home, whenever it works best for you. You can easily create or modify appointments, online or through the app, so that you’re creating a schedule that fits your needs. There are also myriad ways to hold sessions—video chatting, live voice recording, phone calls, in-app messaging, and texting are all offered. Consider the following reviews of BetterHelp counselors from people seeking help with communication and conflict resolution.

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"Pat has been an incredible advocate for me! She checks in and cheers me on and has given me advice and tools to deal with professional and personal/familial conflicts that left me doubting myself. She's been instrumental in helping me discover and unpack learned behavior I wasn't even aware of and helping me understand and establish healthy boundaries with people in my life. I can undoubtedly say that I've been feeling better about myself and more comfortable with the way I walk through the world in large part thanks to her."

"Blaire has been amazing. She's super supportive, empathetic, and kind. She has helped me gain confidence in myself and learn that it is okay to enforce healthy boundaries in my relationships."

Conclusion

Triangulation is, at its core, a manipulation tactic, even if the instigator doesn't realize they're doing it. Trying to involve multiple people in a two-person conflict can end up with everyone losing. If you're having problems with someone, don't try to get others involved. Be someone who can confront the person and have a levelheaded conversation about things.

If you've realized that you're triangulating unintentionally, it's never too late to stop. Be self-aware and mindful about it. As soon as you realize you're having a problem, stop and make sure you're not doing it again. Instead, make sure you're only speaking to the person involved.

It can be difficult, especially if you have a hard time with confrontation. However, direct, honest communication is possible—all you need are the right tools. Take the first step today.

Examples of Narcissistic Triangulation & Effective Responses

Published: December 28, 2021 Updated: September 16, 2022

Published: 12/28/2021 Updated: 09/16/2022

Triangulation is a relational dynamic where two people disagree, and a third person gets pulled into the disagreement; this forms a “triangle” within the argument. ¹ While many people don’t do this intentionally or maliciously, in the hands of a narcissist triangulation is absolutely an intentional tool of manipulation that is often used to control the situation. While narcissistic triangulation can feel frustrating and disarming, understanding what it is and how to respond can help get you out of this toxic triangle.

If you’ve got a narcissist in your life, get the support of a professional. BetterHelp has over 20,000 licensed therapists who provide convenient and affordable online therapy. BetterHelp starts at $60 per week. Complete a brief questionnaire and get matched with the right therapist for you.

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What is Triangulation?

Triangulation is an emotional manipulation tactic where two people in an argument try to pull in a third person to change the argument dynamic. The goal of triangulation is to divide people within the argument and to tip the scales of the argument. Often this is done with indirect communication, with the third person being the go-between of the two people who are initially disagreeing.

People often use triangulation to:

Distract from the real issue or argument
Tip the scales of the argument in their favor
Reinforce their sense of superiority
Attempt to resolve the argument through the third person
Divert some of the stress from the argument onto the third person

What Is Narcissistic Triangulation?

It’s not uncommon for people to employ triangulation unintentionally, because conflict is difficult and uncomfortable to navigate. However, the piece that makes triangulation from a narcissist different is that it is intentional. This often shows up in consistent patterns, which Dr. Stephen B. Karpman coined as “roles.”²

The three roles of triangulation are:²

The Persecutor: The narcissistic person in this role is typically seen as the instigator. The person in this role might use threats, blame, criticism, personal attacks or harsh/aggressive language in order to establish who is the “wrongdoer” in their eyes.
The Victim: When a narcissist is in this role, they will present themselves as helpless, being taken advantage of, and in need of someone to rescue them. By doing this, they are able to deflect responsibility for their own words and actions onto someone else.
The Rescuer: This role typically falls to a support person in the narcissist’s life, but sometimes a narcissist will engage in this role to gain more of a sense of superiority in this way as well. The person in this role fixes things, “smooths things over”, and sometimes accepts responsibility for the narcissist’s feelings or behaviors in order to restore peace.

Why Do Narcissists Triangulate?

Narcissists have a very fragile self-esteem, and they need a constant supply of attention and admiration in order to cope, otherwise known as filling their narcissistic supply. Engaging other people in triangulation fills narcissistic supply in several ways.

The first way that triangulation fill’s narcissistic supply is by increasing their feelings of superiority.³ By bringing in a third person to agree with them and make the other person look wrong, it increases their sense of “rightness” and superiority over the other person.

Triangulation also is a way for narcissists to gain attention, particularly when they’re in the “victim” role. When a narcissist takes on this role, they’re looking for the “rescuer” in the triangle to reassure them, dote on them, and make up for the “persecutor’s” actions that threaten the narcissist’s fragile ego, causing narcissistic injury. Ultimately, all of these things provide the narcissist with what they desire most: attention, affection and reassurance.

Lastly, the cornerstone of narcissistic personality disorder is a deep need for control. If a narcissist is not getting what they want (i.e. winning the argument on their own using manipulation) they will change the dynamic from two people to three. By bringing in a third person and manipulating the third person’s view of the situation to mirror their own, the scales of the argument inevitably get tipped in the narcissist’s favor.

By changing the odds from one-on-one to two-on-one, the disagreeing party often feels ganged up on and outnumbered, and will disengage from the argument. This leaves the narcissist feeling like they “won” the argument, which fills their desire to feel right and superior.

5 Examples of Narcissistic Triangulation

Narcissistic triangulation can and will show up in any relationships within that person’s life, particularly within relationships that the narcissist cannot avoid (for example: family ties do not change, but a narcissist can stop being friends with people once their control is restricted) and in any and all relationships where their sense of control feels threatened.

1. Triangulation With Parents & Children

A narcissistic parent will often use their children in triangulation when their partner pulls away, disengages from the relationship, or altogether leaves the relationship. The narcissistic parent works to buy the child’s love in order to make the other parent look and/or feel bad, and to try to force the other parent to revert to doing things in the way the narcissistic parent thinks they should be done (i.e. giving back all the control to the narcissistic parent).

Triangulation between a parent and children might look like:

Offering treats or indulgences the other parent doesn’t normally allow
Convincing the child that the parent who left is solely at fault
Ignoring the other parent’s reasonable rules and limits, and setting unreasonable rules and limits
Refusing to communicate directly with the other parent, and instead only passing along information through the child
Parents venting and expressing their emotions about the other parent to their children, in order to influence the child’s feelings about the other parent
The narcissistic parent asking about the other parent’s life through the child, in order to use that information later as leverage against the other parent

2.

Narcissistic Triangulation Between Siblings

Another common way that narcissistic parents triangulate within their families is by triangulating siblings against each other. The parent will do this in order to feel like they have control within their family, and to encourage a child to seek out the narcissistic parent’s approval.

Triangulation between siblings often shows up as:

The parent designating one child as a “perfect” or “favorite” child, while designating the other sibling as a “problem child” who should alter their behavior to gain the parent’s approval
Alternating what child receives the parent’s focus of affection and approval in order to make the siblings feel they’re in competition for the parent’s love

This could also happen with a narcissistic sibling pitting a parent against another sibling, or vice versa.

3. In Romantic Relationships

Narcissists in romantic relationships will often use triangulation in order to reassure themselves of the partner’s affection and devotion, as well as to ensure they maintain control within the relationship. ⁴

Some ways that narcissistic triangulation can come into a romantic relationship might be:

Sharing with you that their ex won’t leave them alone or wants to get back together with them in order to get you to give them reassurances or behave in certain ways that fill their narcissistic supply
They might call your mother and complain how badly you’re mistreating them or being unfair, in order for you to be reprimanded by her later
Bringing a platonic friend into one of your arguments and asking that friend to choose a side (typically a narcissist will prime that friend to influence them to choose the narcissist’s side)

4. Your Boss Uses Narcissistic Triangulation

Dealing with a narcissistic boss is a particularly tricky situation because the boss already holds more power than the employee, which impacts your paycheck and your overall life quality.⁵

Some ways that a narcissistic boss might use triangulation could look like include:

Publicly comparing an employee that isn’t as productive to a team member that is more productive
The boss confiding in you about potentially having to hire a weaker employee, in order to influence your view of your boss or to alter your behavior in some way
Asking for your input to make a final business decision that two employees are disagreeing about, instead of making the decision as the boss

5.

Narcissistic Triangulation Between Coworkers

Narcissistic triangulation often shows up in the workplace, since it offers a passive aggressive way to even the playing field with workplace rivals and manipulate interactions between coworkers.

Some ways narcissistic triangulation shows up in the workplace between coworkers include:

A coworker sharing with the boss that you don’t deserve your leadership role, in order to secure that position for themselves
Gossiping to you that someone else treated them poorly, in order to have you stand up for them in the future
After disagreeing with a narcissistic coworker, they might triangulate the boss into the situation in order to get you reprimanded as a ramification for disagreeing with them

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How to React to Narcissistic Triangulation

Once you’re able to recognize and understand the dynamics of triangulation, the next step is to figure out if you want to change the dynamic and how to go about doing so. Changing and avoiding triangulation is often difficult, and narcissistic individuals will most likely fight the change every step of the way.

Here are seven strategies that can help you respond healthily to narcissistic triangulation:

1. Recognize When Triangulation Is Happening

The first step to shifting this pattern is to work on recognizing when you’re being pulled into a triangle dynamic. Understanding that this is a manipulation tool can help you release some of the guilt they’re putting on you.

Review interactions with people who engage you in triangulation afterward with yourself, with a trusted individual outside of the triangle, or with a therapist. Even if you don’t notice the triangulation until after it’s already happened, you’re more likely to recognize it sooner in the future.

2. Work Toward Accepting Your Role in the Triangulation

It’s likely that you have played all the roles of a triangulation dynamic at some point when you’re in a relationship with a narcissistic person. There might have been times you were aware of your role, as well as times you weren’t fully aware of this dynamic. Acknowledging and accepting what you have brought into the triangulation is the first step in changing this relationship pattern.

Again, remember that this is a manipulation tool, and people with narcissistic traits are master manipulators. You cannot change the past, but you do get to decide how you engage in the future — including what roles of a triangle you do or do not decide to engage in.

3. Have a Direct, Private Conversation

Highlighting triangulation in a direct way can feel stressful, especially when one person is intentionally sowing division. Remember that a narcissist is engaging people in triangles with them in order to gain control of the person or situation. Speaking to them privately about their behavior removes some of the possibility that they will blow up in order to seek approval and defense from onlookers, but it also lets the narcissist clearly and definitively know that you aren’t willing to continue to engage in this dynamic.

Essentially, you’re letting them know that they won’t be able to control your emotions and reactions moving forward. They may think twice before attempting to engage you in triangulation again. Please note that sometimes a narcissist will still test this to make sure your actions will follow through with your words, so being prepared for that is your next course of action.

4. Respond, Don’t React

Sometimes having a private and direct conversation is not enough to make a narcissist stop trying to pull you into a relational triangle. Remember that in these situations, the narcissistic person is seeking your emotional reaction in order to draw you into your typical patterns.

In order to reinforce the idea that you will not be engaging in triangulation anymore, you just have to not engage. In this case, meeting triangulation with silence is actually the loudest way you can communicate this boundary. Using other techniques like “grounding,” learning and practicing phrases to disarm a narcissist, or mentally rehearsing this situation beforehand with a friend or therapist can also help you hold this boundary in the moment.

5. Set Your Own Boundaries

You may have gotten yourself out of actively participating in narcissistic triangulation, but that doesn’t mean they won’t still talk about you to other people in your life. Setting boundaries with someone you have to see at work or in family settings can be difficult, but you can set personal boundaries that the narcissist cannot directly influence in order to help you navigate the situation.

Boundaries to set with a narcissist include:

Leaving situations where you find yourself alone with them
Ignoring comments that are meant to bait you
Avoiding sharing personal information
Not going into situations without certain trusted people present who understand the situation and will support you

6. Strengthen Your Own Supports

Narcissistic triangulation can severely impact your own self-esteem and emotional well being. Making sure you have your own support outside of this person is crucial in being able to navigate this situation and to stay emotionally and mentally healthy.

Work on intentionally cultivating a network of supportive friends and community — this could be friends, family, neighbors, or people who share similar interests as you. Anybody who is supportive and will not pull you into unhealthy triangles or be judgmental fits this category.

7. Find a Therapist

Having a therapist in your corner is an invaluable tool in navigating narcissistic triangulation and narcissistic abuse. A therapist’s purpose is to help you recognize relational patterns and to help you figure out how to navigate or change those patterns in a nonjudgmental way. Choosing a therapist who can help you recognize and recover from narcissistic abuse is a good place to start. A therapist directory is a great place to start looking.

Final Thoughts

Dealing with a narcissist who is using triangulation can be a stressful and exhausting process, but you’re not alone. Talking to a therapist or reaching out to a trusted friend or family member can make a big difference in how you feel, and can help you to set the boundaries you need to live a healthy and balanced life.

Additional Resources

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BetterHelp (Online Therapy) – BetterHelp has over 20,000 licensed therapists who provide convenient and affordable online therapy. BetterHelp starts at $60 per week. Complete a brief questionnaire and get matched with the right therapist for you. Get Started

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For Further Reading

Mental Health America
National Alliance on Mental Health
MentalHealth.gov

5 sources

Choosing Therapy strives to provide our readers with mental health content that is accurate and actionable. We have high standards for what can be cited within our articles. Acceptable sources include government agencies, universities and colleges, scholarly journals, industry and professional associations, and other high-integrity sources of mental health journalism. Learn more by reviewing our full editorial policy.

Gale J., Muruthi B. (2017) Triangles and Triangulation in Family Systems Theory. Lebow J., Chambers A., Breunlin D. (eds) Encyclopedia of Couple and Family Therapy. Springer, Cham. https://doi.org/10.1007/978-3-319-15877-8_758-1
Karpman, S. (1968). Fairy tales and script drama analysis. Transactional Analysis Bulletin. 7(26), 39-43. https://karpmandramatriangle.com/pdf/DramaTriangle.pdf
American Psychiatric Association. (2013). Diagnostic and statistical manual of mental disorders (5th ed.). Arlington, VA.
Back, M. D., Schmukle, S. C., & Egloff, B. (2010). Why are narcissists so charming at first sight? Decoding the narcissism–popularity link at zero acquaintance. Journal of Personality and Social Psychology. 98, 132–145. https://pubmed.ncbi.nlm.nih.gov/20053038/
Germain ML. (2018) Narcissism in Leadership and Management: A Research Summary. Narcissism at Work. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-60330-8_4

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Delaunay triangulation algorithm using the sweeping line method / Sudo Null IT News

Good day!

In this article I will describe in detail the algorithm that I got as a result of using the idea of a “sweeping line” to construct a Delaunay triangulation on a plane. It has several ideas that I have not seen anywhere when reading articles about triangulation.
Perhaps someone will also find them unusual. I will try to do everything in the best traditions and include the following things in the story: a description of the data structures used, a description of the steps of the algorithm, a proof of correctness, time estimates, and also a comparison with an iterative algorithm using a kD-tree.

Definitions and problem statement

Triangulation

It is said that a triangulation is given on a set of points in a plane if some pairs of points are connected by an edge, any finite face in the resulting graph forms a triangle, the edges do not intersect, and the graph has a maximum number of edges.

Delaunay triangulation

A Delaunay triangulation is such a triangulation in which for any triangle it is true that there are no points from the original set inside the circumscribed circle around it.

Note : for a given set of points where no 4 points are on the same circle, there is exactly one Delaunay triangulation.

Delaunay condition

Let triangulation be given on a set of points. We will say that some subset of points satisfies the Delaunay condition if the triangulation restricted to this subset is the Delaunay triangulation for it.

Delaunay triangulation criterion

The fulfillment of the Delaunay condition for all points that form a quadrilateral in a triangulation is equivalent to the fact that this triangulation is a Delaunay triangulation.

Note : for non-convex quadrilaterals the Delaunay condition is always satisfied, and for convex quadrilaterals (whose vertices do not lie on the same circle) there are exactly 2 possible triangulations (one of which is a Delaunay triangulation).

The task is to construct a Delaunay triangulation for a given set of points.

Description of the algorithm

Visible points and visible edges

Let the minimal convex hull (hereinafter MBO) of a finite set of points (edges connecting some of the points so that they form a polygon containing all points of the set) and a point A lying outside the hull be given. Then a point of the plane is called visible for point A if the segment connecting it with point A does not intersect the MVO.

An MVO edge is called visible to point A if its ends are visible to A.

In the following picture, the edges visible for the red point are marked in red:

Note : the Delaunay triangulation contour is the MBO for the points on which it is built.

Note 2 : in the algorithm, the edges visible for the added point A form a chain, that is, several consecutive MVO edges

Storage of triangulation in memory

There are some standard methods, well described in Skvortsov's book [1]. Due to the specifics of the algorithm, I will offer my own version. Since we want to check 4-gons for the Delaunay condition, we will consider their structure. Each quad in the triangulation is 2 triangles that have a common edge. Each edge has exactly 2 triangles adjacent to it. Thus, each quadrilateral in a triangulation is generated by an edge and two vertices opposite the edge in adjacent triangles.
Since two triangles and their adjacency are restored by an edge and two vertices, we can restore triangulation by all such structures. Accordingly, it is proposed to store an edge with two vertices in the set and perform a search along the edge (an ordered pair of vertices).

Algorithm

The idea behind the sweeping line is that all points are sorted in the same direction and then processed one by one.

Let's sort all the points along some straight line (for simplicity, by the coordinate ).
Let's build a triangle on the first 3 points.
Further, for each next point, we will perform steps that preserve the invariant that there is a Delaunay triangulation for already added points and, accordingly, MVO for them.
Add triangles formed by the visible edges and the point itself (that is, add edges from the point in question to all ends of the visible edges).
Check for the Delaunay condition all quadrilaterals generated by visible edges. If the condition is not met somewhere, then we rebuild the triangulation in the quadrangle (I remind you that there are only two of them) and recursively run a check for the quadrangles generated by the edges of the current quadrangle (because only in them, after the change, the Delaunay condition could be violated).

Note : in step (4), when starting recursively, it is possible not to check the quadrangles generated by the edges emanating from the point considered at this iteration (there are always two of four). Most often they will be non-convex, for convex the proof is purely geometric, I leave it to the reader. Further, we will assume that only 2 recursive runs are performed for each rebuild.

Delaunay test

Methods for checking quadrilaterals for the Delaunay condition can be found in the same book [1]. I will only note that when choosing a method with trigonometric functions from there, with inaccurate implementation, negative values of the sines can be obtained, it makes sense to take them modulo.

Search for visible edges

It remains to understand how to efficiently find visible edges. Note that the previously added point S is in the MVO at the current iteration, since it has the largest coordinate , and is also visible to the current point. Then, noticing that the ends of the visible edges form a continuous chain of visible points, we can go from the point S in both directions along the MBO and collect the edges while they are visible (the visibility of the edge is checked using the cross product). Thus, it is convenient to store the MVO as a doubly linked list, at each iteration removing visible edges and adding 2 new ones from the considered point.

Visualization of the algorithm

Two red dots - added and previous. The red edges at each moment make up the recursion stack from step (4):

Correctness of the algorithm

To prove the correctness of the algorithm, it suffices to prove the conservation of the invariant in steps (3) and (4).

Step (3)

After step (3), obviously, some triangulation of the current set of points will be obtained.

Step (4)

During the execution of step (4), all quadrangles that do not satisfy the Delaunay condition are on the recursion stack (follows from the description), which means that at the end of step (4), all quadrangles satisfy the Delaunay condition, that is, the Delaunay triangulation has actually been built. Then it remains to prove that the process in step (4) will ever end. This follows from the fact that all edges added during rebuilding come from the current considered vertex (that is, there are no more than ) at the step) and from the fact that after adding these edges we will not consider the quadrangles generated by them (see the previous remark), which means that we add at most once.

Time complexity

On average, on uniform, normal distributions, the algorithm works quite well (the results are shown in the table below). There is an assumption that the time of its operation is . In the worst case, the estimate takes place.

Let's break down the running time by parts and understand which one has the biggest impact on the total time:

Sort by direction

For sorting, we will use the score.

Search for visible edges

First, let's show that the total time spent searching for visible edges is . Note that at each iteration we find all visible edges and 2 more (the first not visible) in linear time. In step (3) we add 2 new edges to the MVO. Thus, in total, there will be no more than edges in the MVO that changes throughout the algorithm, which means that there will be no more than different visible edges. We will also find edges that are not visible. Thus, there are at most edges in total, which corresponds to the time .

Construction of new triangles

The total time for constructing triangles from step (3) with already found visible edges is obviously .

Triangulation rebuilding

It remains to deal with step (4). First, note that checking the Delaunay condition and rebuilding if it fails are quite expensive (although they work in ). Just checking the Delaunay condition can take about 28 arithmetic operations. Let's look at the average number of rebuilds during this step. Practical results on some distributions are given below. According to them, I really want to say that the average number of rebuilds is growing at a logarithmic rate, but we will leave this as just an assumption.

Here we also want to note that the direction along which sorting is performed can greatly vary the average number of rebuilds per point. So on a million evenly distributed on a long, low rectangle with an aspect ratio of 100,000:1, this number varies from 1.2 to 24 (these values are achieved by sorting the data horizontally and vertically, respectively). Therefore, I see the point in choosing the sort direction arbitrarily (in this example, with an arbitrary choice, on average, about 2 rebuilds were obtained) or choose it manually if the data is known in advance.

Thus, most of the program's running time usually goes to step (4). If it is performed quickly, then it makes sense to think about speeding up sorting.

Worst case

In the worst case, at the -th iteration, a recursive call occurs in step (4), that is, summing over all i, we obtain the worst case asymptotics . The following picture illustrates a beautiful example on which the program can run for a long time (1100 rebuilds on average when adding a new point with input data of 10000 points).

Comparison with the iterative algorithm for constructing Delaunay triangulation using kD-tree

Description of the iterative algorithm

I will briefly describe the above algorithm. When the next point arrives, we use the kD-tree (I advise you to read about it somewhere if you don’t know) and find a triangle that is already built quite close to it. Then, by going in depth, we are looking for a triangle in which the point itself falls. We complete the edges at the vertices of the found triangle and actually perform step (4) from our algorithm for new quadrangles. Since the point may be outside the triangulation, for simplicity, it is proposed to cover all the points with a large triangle (build it in advance), this will solve the problem.

Similarity of algorithms

In fact, if the points are added in order sorted by direction, then our algorithm actually works the same as iterative, except that the number of rebuilds is less. The following animation demonstrates this perfectly. On it, points were added from right to left, and all of them are covered by a large triangle, which is subsequently removed.

Algorithm differences

In the iterative algorithm, the localization of a point (search for the desired triangle) occurs on average for , on the above distributions, on average, 3 rebuildings occur (as shown in [1]), provided that the points are supplied in an arbitrary order. Thus, the sweeping straight line gains time from the iterative algorithm in localization, but loses it in rebuilds (which, let me remind you, are quite difficult). In addition, the iterative algorithm works online, which is also its distinguishing feature.

Conclusion

Here I will just show some interesting triangulations resulting from the algorithm.

Beautiful pattern

Normal distribution, 1000 points

Uniform distribution, 1000 points

Tablegulation, built on the locations of all cities of Russia

Here you can see an example of my code of this algorithm:
github.com/pkositsyn/delaunay-triangulation-algorithm

Thank you for your attention!

Literature

[1] Skvortsov A.V. Delaunay triangulation and its application. - Tomsk: Publishing House Vol. un-ta, 2002. - 128 p. ISBN 5-7511-1501-5

Triangulation in family psychology

What is triangulation

Definition 1

Triangulation is a tactic used by manipulating people that involves three people (as its name suggests).

The phenomenon of triangulation is described by M. Bowen. In his opinion, triangulation is the involvement of a third family member in dyadic relations at a time when the increase in anxiety indicates the possibility of manifestation and (or) awareness of the conflict in the dyad or its transition into an uncontrollable form.

The problem with triangulation is that it is used for a conversation intended for two people. Imagine that person A wants to communicate with person B. Instead of communicating directly with person B, he will instead communicate with person C and use person C as a relay. This allows person A to control the communication he has.

Three people can be triangulated:

Victim: This label describes a person who in some situations (if not all) looks like "poor me". Although there are cases where a person is a victim because of how they are treated by others.
Persecutor: This label describes a person who attacks others who are members of the triangle.
Rescuer: This is a person inside or outside the created triangle who acts as a rescuer.

A person who is a pathological liar often reproduces the triangle in many areas of life, including the workplace, school, neighborhood, and home environment. The triangle is sometimes deliberately created to create chaos and thus give the person the opportunity to appear as a victim.

Triangulation is a tactic that people use for many reasons. It can be used to compete or to play people against each other. If a person is narcissistic and manipulative, they may use triangulation to control a conversation or communication. Let's say you know someone who always lies. How can this liar gain trust? For example, he might tell different friends (or family members) the same lies he told you, and then clarify that mutual friends believe him. You call your friends and ask them about it. They, in turn, may be upset because of such a hostile attitude towards a common friend. People don't like being interrogated, and no one wants to get involved in a fight between two people. As a result, a triangle is formed, which is difficult to destroy.

Triangulation is a concept that therapists and other professionals study to understand how certain traumatized people operate in interpersonal relationships. Many people who have experienced traumatic situations in their lives (abuse, car accident, fire, natural disaster, loss of home, death of a loved one, neglect by parents, etc.) perceive life and relationships in a slightly different way. They are influenced by social and emotional personal experiences, the environment, how others treat the person, etc.

People who are typically abused and neglected in childhood are often involved in triangulation either intentionally or unintentionally. This is a very complex process that negatively affects all participants. In fact, many people with personality disorders (borderline personality disorder, antisocial personality disorder, sociopathy), mood disorders (bipolar disorder, depression, etc.), anxiety, or attachment problems may display similar behaviors.

Types of triangulation

For many families or friends, triangulation becomes an ongoing or chronic process in which a person participates either to achieve an end goal (meeting one's needs, controlling others, gaining attention, etc.) or to hide the truth about a situation .

If a pathological liar is in a state of anger and resentment, he can manipulate the situation in such a way that the other member of this triangle looks like an "unstable" person. This happens in many cases of triangulation. For example, during a divorce, resolving issues with custody of a child, in conflict relations between parents and children, etc.

As hard as it is to believe, some people inadvertently triangulate and do not understand what they are doing to others. This type of person includes: a passive-aggressive person, a shy person, someone who hasn't "found their voice", or someone who is afraid (or doesn't know how) to be assertive. Other people simply lack understanding of their behavior.

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