• # A Brief Introduction to Lerp

General and Gameplay Programming

Linear interpolation (sometimes called 'lerp' or 'mix') is a really handy function for creative coding, game development and generative art.

The function interpolates within the range [start..end] based on a 't' parameter, where 't' is typically within a [0..1] range.

For example, divide 'loop time' by 'loop duration' and you get a 't' value between 0.0 and 1.0.

Now you can map this 't' value to a new range, such as lerp(20, 50, t) to gradually increase a circle's radius, or lerp(20, 10, t) to gradually decrease its line thickness.

Another example: you can use linear interpolation to smoothly animate from one coordinate to another. Define a start point (x1, y1) and end point (x2, y2), then interpolate the 'x' and 'y' dimensions separately to find the computed point in between.

Or use linear interpolation to spring toward a moving target. Each frame, interpolate from the current value to the target value with a small 't' parameter, such as 0.05.

It's like saying: walk 5% toward the target each frame.

A more advanced example, but built on the same concept, is interpolating from one color (red) to another (blue).

To do this, we interpolate the (R, G, B) or (H, S, L) channels of the color individually, just like we would with a 2D or 3D coordinate.

Another quick example is to choose a random point along a line segment.

There are lots of ways to use linear interpolation, and lots more types of interpolation (cubic, bilinear, etc). These concepts also lead nicely into areas like: curves, splines and parametric equations.

Source code for each of these examples is available here: https://gist.github.com/mattdesl/3675c85a72075557dbb6b9e3e04a53d9

Matt DesLauriers is a creative coder and generative artist based in London. He combines code and emergent systems to make art for the web, print media, and physical installations.

Note:
This brief introduction to lerp was originally published as a Twitter thread and is republished here with the kind permission of the original author.

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## User Feedback

Very cool article!  I use linear interpolation to smooth out visual movement, it works very nicely with variable render rates. It's also nice for visual effects like shown above.

Thanks for sharing!

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Nice article! I loved the visual examples.

I can't say how many times lerp has been the tool of choice to achieve zillions of nice looking effects using shaders, plus is the basis of rasterization!

Keep the good work

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Note that your "springing toward" example (lerping a fraction of the remaining distance each time) is highly frame-rate dependent.  So if your game isn't frame-rate locked (i.e. web-games running on machines with different monitor refresh rates, or even a locked 60 fps game running on a machine too slow to support it) different people may see different behaviors.  The spaceship rotation in George Prosser's DRILL_BIT, for instance, is much harder to use on a slow machine.

And the way you're doing it, scaling by the deltaTime, will only partly correct for that.  For instance, if your framerate is twice as fast (deltaTime is half the length) then you actually need the square root of the rate (you lerp twice in the same time, essentially multiplying by the factor twice, i.e. squaring it).  In the general case you need Math.pow:

// Return a lerp factor for the given frame time which will result in
// the value getting convergenceFraction of the way to the target in
// smoothTime time units.  I usually just use 0.9 or 0.95 for the fraction
// and vary the smoothTime to get the effect I want.
function smoothOver(dt, smoothTime, convergenceFraction) {
return 1 - Math.pow(1 - convergenceFraction, dt / smoothTime)
}

For further adventures in non-linear interpolation things, Squirrel Eiserloh's Fast and Funky 1D Non-linear Transformations GDC talk is well worth a watch.  Although note that with his later (curved) functions, he only mumbles once that he's "normalizing" them (meaning scaling them up so they are exactly one unit tall).  Without knowing that, if you try to use some of the formulas he gives as written, you'll end up with something very flat that does almost nothing.  But he gives a good intuitive introduction to how to think about constructing some of these things.

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• By Ey-Lord
Hello everyone

I am here to gather your opinion, remarks, ideas or any constructive criticism you may have about what I am going to present. Don’t be shy!

A bit of background:

I am working alone on an indy web-based game, a simulation of RPG (idle game) where the player controls a group of 4 characters that he can sent into battle and that will fight automatically based on some AI preference that are similar to the FF 12 system (but more complex / powerful). He then earns some experience and resources that he can use to improve his unit’s gear, talents and skills. He has a lot of control on what skills his characters will use and how/when.

What brings me here today:

The AI of Monsters. I have the AI settings for players covered (basically a bunch of if/then/and/or/else settings that he can combine and order so that his units will act as he intends in battle). I’ve been working on the AI of monsters for quite some time, made a long break and came back recently to it.

Short description of the battle system:

No movement involved. Battle is fully automated. Players setup its units AI settings before battle and monsters are controlled by a separate AI. This is a 4v4 battle, like FF7 with some kind of ATB and any time a unit fill its ATB, it can play and the then the next unit who will fill it will play, etc. The player is completely free of his playstyle and may create very offensive group or very defensive ones. 4 healers or 4 tanks is completely possible.

The battle system is very complex and allows for very varied and sometimes unusual strategies, like killing your own allies to proc an “on death buff” that will be devastating for the opponent.

What I want for my AI?

It needs to be fun to fight against and challenging. Ideally, I would like an AI as smart as possible (not omniscient but thinking as a human would). I know that a super-smart AI is not always the best way to make a game fun or challenging but in the context of my game, this is the result I want to achieve. It may seem unfair to have the AI try to kill your squishy while your tank is standing right there but my class design gives the tools to players to counter that so it’s not an issue (tanks are not purely aggro based for example). I want players to always be challenged by AI moves and that they must carefully think about their strategy because if they leave a big hole in it, I want the AI to exploit it.

In practice, it means a few requirements:

No dumb decision / do not fall into obvious player’s traps Exploit obvious flaws of the opponent Act in coordination when appropriate with other units Able to find who should be their focus in the player’s team (some notion of threat) Find the best move to use and if there is some kind of combo possible, use it

These requirements are harder to meet than it looks. The issue is the sheer number of different mechanisms and strategies available to players and to monsters as well. For example, there are many cases where killing or attacking a player unit might be detrimental (units that return damages or that gain power when you hit then for example).

What I have tried before?

I have tried or at least reviewed many different AI concepts so far.

-          A simple copy of my player’s AI system (hierarchical if/then/else). It was easy to script as I already have the UI in place for players so I can quickly define a basic AI for any new monster’s group. The main drawbacks are that it needs to be written for every monster group, it does not allow smart targeting and cannot find the best target or the best skill to use. It will also make dumbs decision as the targeting options cannot assess threats at all.

I’ve rules out planners since for purely selecting the best pair of (skill, target), they do not seem to match my needs.           (H)FSM or BT does not seems to match my needs as monsters do not have states / transition condition that can lead to something useful for me.        I’ve ruled out aNNs as they might, with proper training, be able to find the best action at a given time but it’s very tedious to implement and will not solve my need of finding combo or coordinating with other units very well. (plus, let’s be honest, I’d be a bit out of my depth to program them)           I have spent an extensive period of time trying with tree searches. Mainly: monte-carlo with random sampling and came to the conclusion that due to the complexity of my battle system, it is excessively costly to compute any kind of reliable data this way.
-        My current AI system is a version of my first one (the same as the players) but with access to some “smarter” targeting function that in theory allow to choose the best target. These functions work by gathering data for thousands of simulated fights during the AI time to play (1 second). It’s a first step to find the best target but not very accurate (lots of big flaws that can be exploited by players) and it is very time consuming and that is something I’m trying to get away from. I do not want to use 100% of the players CPU as I do now.

What is my latest idea?

I started to study more in-depth the Utility theory as described by Dave Marks (I read his book and watched his GDC AI lectures as well). I liked the idea. I like that I can start on something relatively simple and add more considerations as things progress to handle more and more situations. While my work began as something very close to utility theory, it evolved a bit afterward. Here is what I plan on doing to compute a unit’s best course of action:

A – Score every of its move (each move is a pair [skill, target]).

B – Chose the move according to a selection strategy (highest score, weighted random, random amongst the top scores… lots of different selection algorithm can be used there).

So far, easy, right? Let’s dig deeper into our first phase of scoring (A), which is the hard part. For all the damage or healing skills:

Step 1: The final scoring of the move [skill,target] will be function of the a “Survival” scoring for the player team and for the enemy team. An example of this relationship could be: Adding all the survival scores of each unit in Team A and divide the result by the addition of all the survival scores for each unit in team B.

Step 2: The survival score of each unit will be its Health after the move we are evaluating, divided by the total damage per turn that we estimate other units can deal to her (minus the total heal it ca receive). [This a step where we can process damage and heal over time as well]

Step 3: This damage per turn estimation will be, initially, the sum for every unit in battle of the damage or heal per second it can deal to that unit. For example: If I’m alone vs 2 bad guy that can deal 1 dmg/turn and if I can deal 1 heal/turn, the damage per turn estimation against me will be 2-1 = 1. [This is not optimal since we are counting the damage of each unit once per enemy unit but it’s a start]

Step 4: To compute the DPS or HPS of each unit, we review the unit’s skills and compute their output against the unit we want to evaluate it against. From that, we construct a skill sequence to maximize the damage output and once we got the optimal skill sequence, we can compute its DPS or HPS output and pass it along for Step 3.

It might seem like a lot of work, since, in a world with only damage or healing skills, the DPS or HPS sequence of each unit will be the same in every situation and as such only the damage done or healing done by the skill evaluated would be enough. But…

The tricky part comes from buffs and debuffs. If we use the above algorithm, (de)buffs that changes the damage or healing someone does or receive will be evaluated correctly as it will change the damage or heal per second output of units and it would affect the survival score and the final scoring. That is why I chose to include DPS and HPS computations for each unit for each move.

This is all fine until we consider (de)buffs that changes the power of other (de)buffs. Like: I cast a buff that double the length of all my future buffs. My algorithm can’t evaluate it correctly. It’s a situation that will be common enough in my game and I want my AI to deal with it. Note: there are more complex situations where a unit could buff a buff that buffs a buff that buff a buff [….] that will end-up buffing a damage or healing skills, but those cases will not be addressed as they will hopefully be rare and too cumbersome to compute anyway.

So, my goal is to score properly buffs that:

Buffs the damage or healing output of someone           Buffs that buffs a skill that does the above

L    Long story short of how I am doing that. I’m using my initial algorithm but while also estimating damage or healing per second change for each dps or hps sequence.To do that I’m evaluating every move of the unit (or every unit in case of AoE but lets keep it simple with single target) that is targeted by the buff. So, we are switching PoV here compared to the initial unit we are evaluating (unless the move evaluated is buffing itself)

-          I’m doing the above in 2 situations:

o   A : After a cast of the buff skill I’m evaluating

o   B : Without the cast of the buff, just like if it was that unit’s turn to play

-          Using a sort of min/max approach: if the unit targeted by the buff is an ally, we will take the best branch of our tree in A and compare it with the same branch (pair [skill,target]) in B. If the unit targeted by the buff is an enemy, we want to lower their maximum score and will select the tree branch that does that in A to also compare it with the same branch in B.

-          The information we extract here are DPS or HPS delta for each sequence of DPS/HPS for each unit vs each other unit.

-          Then, we go back to our steps 1 to 5 and compute our scoring for the move (buff) while using our new dps/hps deltas to get better and more accurate dps/hps sequence for units affected by the buff.

This is basically it. I’ve ran a manual version of the algorithm in 2 different battle settings to test it and see if it gave good results. It worked. Not flawlessly but it worked. Lots of cases will still require tweak and additions to the basic idea but I think its promising. (taunts and CCs are not easy to deal with but it’s manageable)

What I like is that I can add more considerations later (as in the utility theory) like: resource cost, general unit strategy (cleave or focus), behavior (careful, lunatic, reckless). While this will still be a bit time consuming it should be a good order of magnitude faster than my current AI. It also does not prevent me from adding hardcoded AI move if I want to “script” more some monsters. Debugging and tweaking might be a bit painful though, especially when fights will involve lots of skills & stats but that’s an issue that most AI for my game would likely have anyway.

To come back with my initial goals:

No dumb decision / do not fall into obvious player’s traps
o   Not perfect but it should choose the best target whenever possible

Exploit obvious flaws of the opponent
o   Same as above

Act in coordination when appropriate with other units
o   This can be done simply by adding weight to some targets or computing moves for all units of a group before deciding which one to take (for example to take the best move vs a specific unit, on average)

Able to find who should be their focus in the player’s team (some notion of threat)
o   It will naturally focus the unit who is the easiest to kill and debuff or CC the ones that deal the more heal/damage. But, to better solve this, we will need to add other considerations to the AI scoring process, It should not be *too* hard

Find the best move to use and if there is some kind of combo possible, use it
o   Combo are very often in the form of buff/debuff setup before an actual damaging or healing skills and my AI can compute up to a 3 moves combo (buff > buff > skill that dmg or heal) which should cover most cases.

I’m quite happy with my initial tests. I’m not going to be coding it now. My goal was to reflect on the subject on paper and try to see if designing my AI would be a roadblock or not for my project. There are a few other area I want to design and take time to really think about before getting back to my project full time. I’d love to hear your toughs and feedbacks about my AI ideas. Do you see huge roadblocks I’m missing? Does it sound ok to you?

If you read that far…. thank you and I can"t wait to hear from you guys😊

• By Sword7
I googled for terrain collisions but found only some sources about flat worlds.   I want collision detection for spherical terrain worlds.   I have "3D Engine Design for Virtual Globe" book but it did not mention any algorithms about collision detection for landing, rolling, walking, crashing, etc...   Does anyone have any good sources for spherical terrain collision detection?
• By bzt
Hi guys,

I've an OpenGL question, which is quite math ad linear algebra related.
Let's assume we have two coordinate systems, S (scene) and O (object). I'd like to place O inside S, so I need O' (in S coordinates). Using the following transformation matrices I can do that: rotation, scale, displacement. So far so good. I have two questions though:

1) assuming the place of O' is specified with 4 points (zerus, and one for each axii unit vector end points) how can I calculate the required transformation matrices?
It's a "simple" case, as let's say points are P0, P1, P2, P3 and x = P0->P1, y = P0->P2, z = P0->P3. Also |x| = |y| = |z| (all has the same length) and they enclose 90 degree with each other. This surely can be solved using standard GL transformations easily, I just need an algorithm to calculate the matrices from P0, P1, P2, P3.

2) the more difficult question, how can I do the same if O' can be distorted, so |x| != |y| != |z| and their angle is not necessarily 90 degree? (Imagine that you "sit" on O, so O' became stretched and it's top became larger and moved to the side, so that angle(x, y) = 80 degree for example). How to get the required transformation matrices in this universal case, when you only know P0, P1, P2, P3?

Hope it's clear what I'm asking. I need an algorithm to generate a transformation matrix that I can then use to transform all points in O into O'.
bzt

• I'm working on a step-time strategy game (not sure if that's the proper term, but it moves in discrete time steps, about 4 frames, before pausing for input) with positively monstrous pathfinding calculations. I've solved the biggest of my problems already, but there is one algorithm that causes significant slowdown, and I'm sure it could be optimized, but I'm not sure how best to do that because my case has a unique property that isn't covered in any A* optimizations I've researched. (Or perhaps I'm just not thinking about it the right way.)
My game allows the creation of formations, which are defined as positional groupings of units relative to a leader. It's extremely important to the gameplay that these formations keep their shape as best as possible while balancing the need to move smoothly through any narrow spaces that constrict their shape. The algorithm I'm asking about now is responsible for choosing a path for the leader of the formation that (1) avoids unnecessarily traveling into narrow spaces and (2) always allows a certain amount of clearance between itself and obstacles, whenever those things are possible. Where I have found typical clearance-based approaches to A* fall short in addressing my problem is, ultimately, the leader will choose a path that goes through areas even just 1 space wide if there is no other option. In other words, I can't have the algorithm disregard those, but I can't have it explore those nodes either until it has explored the entirety of the rest of the map.
Here is my basic test map for this. The 5 green symbols at the top-left represent a formation led by the V in the center. The yellow circle at the center of the map is the goal.

Here is the solution that a regular A* would give, finding the shortest path:

Here is the solution I want (and the algorithm already does this, just slowly)

Here is the Python code that achieves this -- I know there are going to be some questions about what the vectors are and all of that, but basically, this is just a typical A* setup with the addition of a "c_cost," which is 0 if the given node has all the space that it wants (as defined by optimal_area), or 2 ^ the number of spaces missing from the amount of space we want.
def leader_pathfind(leader, going_to, treat_passable = [], treat_impassable = []): """ Seek the shorest path that attempts to avoid obstacles at a distance equal to that of the furthest member in the formation """ global game_width global game_height global diagonal_movement_cost global formation_dict at_now = leader['position'] treat_passable.append(going_to) # Prevents indefinite hanging while rerouting if an obstacle has moved to the destination if at_now == going_to: return [] formation_obj = formation_dict[leader['id']] vectors = formation_obj.get_formation_vectors() max_dimension = 0 for vector in vectors: for axis in vector: if abs(axis) > max_dimension: max_dimension = abs(axis) optimal_area = basic_pow(max_dimension * 2 + 1, 2) def clear_area(position): """ Check in square rings around the position for obstacles/edges of map, then return the total area that is passable within the square that first hits an obstacle/edge of map """ if not position_is_passable(position, treat_passable = treat_passable, treat_impassable = treat_impassable): return sys.maxint distance = 0 hit_obstacle = False while not hit_obstacle: distance += 1 corner_a = (position[0] - distance, position[1] - distance) corner_b = (position[0] + distance, position[1] + distance) outline = rectline(corner_a, corner_b) for point in outline: if not within_game_bounds(point): hit_obstacle = True break elif not position_is_passable(point, treat_passable = treat_passable, treat_impassable = treat_impassable): hit_obstacle = True break elif distance == max_dimension: # We have all the space we want, no need to compute further hit_obstacle = True break tent_area = rectfill(corner_a, corner_b) area = 0 for point in tent_area: if within_game_bounds(point): if position_is_passable(point, treat_passable = treat_passable, treat_impassable = treat_impassable): # Some stray diagonals may get counted here occasionally, but that should be acceptable for this purpose area += 1 return area def c(position): # c_cost is in terms of clear area relative to the optimal clearance area = clear_area(position) c_cost = 0 if area >= optimal_area else basic_pow(2, abs(area - optimal_area)) # Cost grows exponentially by powers of 2 for each point less than the optimal area return c_cost # Traditional A* # def h(position): return basic_distance(position, going_to) / 100.0 f_dict = {at_now: h(at_now)} g_dict = {at_now: 0} open_set = [at_now] closed_set = [] get_last = {} found_path = False current = at_now while len(open_set) > 0: current = open_set[0] for op in open_set: if op in f_dict.keys(): if f_dict[op] < f_dict[current]: current = op open_set.remove(current) closed_set.append(current) if current == going_to: found_path = True break for neighbor in neighbors(current, treat_passable = treat_passable, treat_impassable = treat_impassable): if neighbor in closed_set: continue if not neighbor in open_set: open_set.append(neighbor) movement_cost = diagonal_movement_cost if diagonally_adjacent(current, neighbor) else 1 tent_g = g_dict[current] + movement_cost tent_f = tent_g + c(neighbor) + h(neighbor) if not neighbor in f_dict.keys(): g_dict[neighbor] = tent_g f_dict[neighbor] = tent_f get_last[neighbor] = current else: if tent_f < f_dict[neighbor]: g_dict[neighbor] = tent_g f_dict[neighbor] = tent_f get_last[neighbor] = current if found_path: path = [going_to] current = going_to while current != at_now: current = get_last[current] path.append(current) path.reverse() path.pop(0) return path else: write_log('leader_pathfind: could not find a path from ' + str(at_now) + ' to ' + str(going_to)) return False Jump Point Search, which I've used elsewhere, relies on the assumption of a uniform-cost grid, so I believe I need to rely on abstracting my map down to a smaller number of abstract nodes, and I'm at a loss for how I would do that for this case. What I've read on using abstract nodes rely on things like identifying entrances/exits from abstract nodes, but that won't be useful if the algorithm chooses a node with acceptable entrance/exit points but lots of obstacles in the middle. Here is an arbitrarily chosen slice of the map with the correct path:

Broken up into abstract 5x5 nodes, I am failing to see a pattern I could rely on to give the algorithm the information it needs, especially since a crucial part of the path rests right on the right edge of the two leftmost abstract nodes. To know that this is the correct path, i.e. has sufficient clearance, it would need to calculate the clearance from that edge, dipping into the nodes in the middle. So I can't choose nodes that will be helpful without already having the information that I need the nodes to speed up the calculation of.
That's as far as I've gotten with my thought process. Can anyone make any suggestions, general or specific? Thanks in advance

• How can you cast a 3D object against the environment? In Unity, you can do Spherecasting, Capsulecasting and Box casting, but I want to cast a cylinder. How does one implement shapecasting?
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