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- BlitzMax provides a set of vector structs for working with 2D, 3D, and 4D coordinates. These
- structs come in three variations for each dimension: #Double ( #SVec2D, #SVec3D, #SVec4D),
- #Float ( #SVec2F, #SVec3F, #SVec4F), and #Int ( #SVec2I, #SVec3I, #SVec4I).
- The choice of struct depends on the precision and coordinate system you require for your application.
- Vectors are used to represent positions, directions, and magnitudes in coordinate systems.
- ## 2D Vectors
- 2D vectors are represented by the #SVec2D, #SVec2F, and #SVec2I structs, each catering to
- different data types: #Double, #Float, and #Int, respectively. These structs consist of two
- components, x and y, which correspond to the horizontal and vertical coordinates in a 2D space.
- They are commonly used in 2D games, graphical applications, and geometric calculations.
- ### 2D Games
- In 2D games, vectors play an essential role in various aspects of game development,
- providing a powerful and efficient way to handle mathematical operations related to game
- objects such as characters, projectiles, and other interactive elements.
- **Positions** : Vectors are used to represent the position of game objects in a 2D space.
- The x and y components of the vector store the object's coordinates, which define its location
- on the screen. By manipulating these coordinates, game objects can be moved, placed, or
- arranged in relation to other objects and the game environment.
- In this example, the playerPos and targetPos are 2D vectors ( #SVec2F) representing the positions of
- the player object and the target object. The player object is moved by manipulating the x and y
- components of the playerPos vector based on the arrow keys pressed. We also check if the player
- object has reached the target object by calculating the distance between their positions using the
- #DistanceTo method. If the player reaches the target, the target object is moved to a new random position
- on the screen.
- ```blitzmax
- SuperStrict
- Framework sdl.sdlrendermax2d
- Import brl.random
- Graphics 800, 600, 0
- Local playerPos:SVec2F = New SVec2F(400, 300)
- Local targetPos:SVec2F = New SVec2F(Rand(50, 750), Rand(50, 550))
- While Not KeyHit(KEY_ESCAPE)
- Cls
- Local x:Float = playerPos.x
- Local y:Float = playerPos.y
- ' Update player position using arrow keys
- If KeyDown(KEY_LEFT) Then x :- 5
- If KeyDown(KEY_RIGHT) Then x :+ 5
- If KeyDown(KEY_UP) Then y :- 5
- If KeyDown(KEY_DOWN) Then y :+ 5
- ' Set playerPos with updated coordinates.
- playerPos = New SVec2F(x, y)
- ' Draw player
- SetColor 0, 255, 0
- DrawOval(playerPos.x - 10, playerPos.y - 10, 20, 20)
- ' Draw target
- SetColor 255, 0, 0
- DrawOval(targetPos.x - 5, targetPos.y - 5, 10, 10)
- ' Check if player reached the target
- If playerPos.DistanceTo(targetPos) < 15
- ' Move target to a new random position
- targetPos = New SVec2F(Rand(50, 750), Rand(50, 550))
- EndIf
- Flip
- Wend
- ```
- **Velocities and Accelerations** : Vectors can represent not only positions but also
- velocities and accelerations. In this context, the x and y components of the vector
- indicate the rate of change of an object's position or velocity, respectively. By adding
- velocity vectors to position vectors, game developers can create smooth and realistic motion
- for characters and other moving objects. Similarly, acceleration vectors can be used to simulate
- the effects of gravity, friction, and other forces on the motion of game objects.
- In this example, the `ballPos`, `ballVel`, and ballAcc are 2D vectors ( #SVec2F) representing the ball's position,
- velocity, and acceleration. We update the ball's position and velocity based on the acceleration
- (gravity) and time elapsed (`deltaTime`). When the ball hits the floor, we reverse its vertical
- velocity and apply a bounce coefficient to simulate the loss of energy during the bounce.
- ```blitzmax
- SuperStrict
- Framework SDL.SDLRenderMax2D
- Graphics 800, 600, 0
- Local ballPos:SVec2F = New SVec2F(400, 100)
- Local ballVel:SVec2F = New SVec2F(0, 0)
- Local ballAcc:SVec2F = New SVec2F(0, 9.81) ' Gravity
- Local ballRadius:Float = 20
- Local floorY:Float = 550
- Local deltaTime:Float = 1.0 / 60.0
- Local bounceCoefficient:Float = 0.8
- While Not KeyHit(KEY_ESCAPE)
- Cls
- ' Update ball velocity and position using acceleration and deltaTime
- ballVel = ballVel + (ballAcc * deltaTime)
- ballPos = ballPos + (ballVel * deltaTime)
- ' Bounce the ball off the floor
- If ballPos.y + ballRadius > floorY And ballVel.y > 0
- ballVel = New SVec2F(ballVel.x, -ballVel.y * bounceCoefficient)
- ballPos = New SVec2F(ballPos.x, floorY - ballRadius)
- EndIf
- ' Draw ball
- SetColor 0, 255, 0
- DrawOval(ballPos.x - ballRadius, ballPos.y - ballRadius, ballRadius * 2, ballRadius * 2)
- ' Draw floor
- SetColor 255, 255, 255
- DrawLine(0, floorY, 800, floorY)
- Flip
- Wend
- ```
- **Collision Detection and Resolution** : Vectors are crucial in detecting and resolving collisions
- between game objects. By comparing the positions and sizes of objects, developers can identify when
- two objects are overlapping or in close proximity, triggering a collision event. Vectors can also
- be used to calculate the response of objects upon collision, such as bouncing, sliding, or sticking
- together. This is achieved by reflecting or altering the velocity vectors of the colliding objects
- based on their properties, such as mass, elasticity, and friction.
- **Distances and Angles** : Vectors can be used to calculate distances and angles between objects,
- which is useful for various game mechanics, such as determining the line of sight for a character
- or measuring the range of a weapon. By subtracting the position vectors of two objects, developers
- can find the vector connecting them and calculate its length to obtain the distance between them.
- Additionally, the dot product and other vector operations can be used to determine the angle between
- two vectors, which is helpful for calculating relative orientations or aiming projectiles.
- In this example, the `characterPos` and `targetPos` are 2D vectors ( #SVec2F) representing the
- character's and target's positions. We calculate the connecting vector by subtracting the
- character's position from the target's position. We then use the `Length()` method to find the
- distance between them and the #ATan2() function to calculate the angle. The distance and angle
- are displayed on the screen.
- ```blitzmax
- SuperStrict
- Framework SDL.SDLRenderMax2D
- Import BRL.Math
- Graphics 800, 600, 0
- Local characterPos:SVec2F = New SVec2F(100, 300)
- Local targetPos:SVec2F = New SVec2F(700, 400)
- Local distance:Float
- Local angle:Float
- While Not KeyHit(KEY_ESCAPE)
- Cls
- ' Calculate the vector connecting the character and the target
- Local connectingVector:SVec2F = targetPos - characterPos
- ' Calculate the distance between the character and the target
- distance = connectingVector.Length()
- ' Calculate the angle between the character and the target
- angle = ATan2(connectingVector.y, connectingVector.x)
- ' Draw character
- SetColor 255, 0, 0
- DrawOval(characterPos.x - 10, characterPos.y - 10, 20, 20)
- ' Draw target
- SetColor 0, 255, 0
- DrawOval(targetPos.x - 10, targetPos.y - 10, 20, 20)
- ' Draw aiming line
- SetColor 255, 255, 255
- DrawLine(characterPos.x, characterPos.y, targetPos.x, targetPos.y)
- ' Display distance and angle
- SetColor 255, 255, 255
- DrawText("Distance: " + distance, 10, 10)
- DrawText("Angle: " + angle, 10, 30)
- Flip
- Wend
- ```
- In summary, 2D vectors are indispensable tools in game development, providing an efficient and
- flexible way to handle a wide range of mathematical operations related to game objects and their
- interactions within the game world.
- ### Graphical Applications
- In graphical applications, 2D vectors play a vital role in representing and manipulating various
- elements in a coordinate system. They enable developers to perform operations such as translation,
- rotation, and scaling, which are essential for rendering and adjusting graphics on the screen.
- **Points** : 2D vectors are used to represent points in a coordinate system, where the x and y
- components store the horizontal and vertical coordinates, respectively. Points are fundamental
- building blocks in graphical applications, as they define the vertices of lines, polygons, and other shapes.
- **Lines and Shapes** : By connecting points using vectors, developers can create lines, curves,
- and shapes in a graphical application. Vectors are used to store the vertices of these shapes
- and calculate properties such as their lengths, areas, and perimeters. Additionally, vector operations
- can be applied to determine intersections, containments, and other spatial relationships between different shapes.
- In this example, we define a function `CalculateLineLength` that takes two 2D points ( #SVec2D) as
- input, representing the start and end points of a line segment. The function calculates the
- connecting vector between the two points and returns its length.
- ```blitzmax
- SuperStrict
- Framework BRL.StandardIO
- Import BRL.Vector
- Local startPoint:SVec2D = New SVec2D(100, 300)
- Local endPoint:SVec2D = New SVec2D(700, 400)
- Local lineLength:Double = CalculateLineLength(startPoint, endPoint)
- Print "Line length: " + lineLength
- Function CalculateLineLength:Double(p1:SVec2D, p2:SVec2D)
- Local connectingVector:SVec2D = p2 - p1
- Return connectingVector.Length()
- End Function
- ```
- **Translation** : Translation is the process of moving a shape or a group of shapes from one location
- to another within the coordinate system. By adding a translation vector to the position vectors of
- the shape's vertices, developers can shift the shape by a specified amount in both horizontal and
- vertical directions. This operation is essential for animating objects, panning scenes, or aligning
- graphical elements.
- **Rotation** : Rotation is the process of rotating a shape or a group of shapes around a specific
- point in the coordinate system. By applying a rotation matrix, which can be derived from trigonometric
- functions or complex numbers, to the position vectors of the shape's vertices, developers can rotate
- the shape by a given angle. This operation is useful for creating effects such as spinning objects,
- rotating scenes, or adjusting the orientation of graphical elements.
- **Scaling** : Scaling is the process of resizing a shape or a group of shapes by a specified factor in
- the coordinate system. By multiplying the position vectors of the shape's vertices by a scaling vector,
- developers can uniformly or non-uniformly scale the shape in horizontal and vertical directions. This
- operation is crucial for zooming in and out, resizing objects, or adapting graphical elements to different
- screen resolutions or aspect ratios.
- In conclusion, 2D vectors are essential in graphical applications, providing a robust and efficient way
- to represent and manipulate various elements within a coordinate system. They enable developers to perform
- operations such as translation, rotation, and scaling, which are crucial for rendering and adjusting
- graphics on the screen.
- ### Geometric Calculations
- Geometric calculations play an essential role in numerous fields, including computer graphics, physics
- simulations, and engineering applications. 2D vectors are invaluable tools for solving problems related
- to distances, intersections, and areas of shapes, as well as performing linear algebra operations like
- dot products and matrix multiplications.
- **Distances** : Calculating distances between points, lines, and shapes is a common task in geometry.
- Using 2D vectors, developers can determine the distance between two points by finding the length of the
- vector representing the difference between their position vectors. Distances from points to lines or other
- shapes can also be computed using vector operations, such as projections and dot products.
- **Intersections** : Detecting intersections between lines, rays, and shapes is essential for various
- applications, including collision detection in games and ray tracing in computer graphics. Using 2D vectors,
- developers can apply algebraic and geometric techniques to determine if and where two elements intersect.
- For example, the intersection point of two lines can be found by solving a system of linear equations formed
- by their parametric equations, which are derived from their direction vectors.
- In this example, we define a `TLine` type that stores the start and end points of a line segment.
- The function `CalculateIntersection` takes two TLine objects as input and calculates their intersection
- point, if it exists. If the lines are parallel, the function returns #False.
- ```blitzmax
- SuperStrict
- Framework BRL.StandardIO
- Import BRL.Vector
- Local line1:TLine = New TLine(New SVec2D(100, 100), New SVec2D(700, 400))
- Local line2:TLine = New TLine(New SVec2D(400, 100), New SVec2D(200, 600))
- Local intersection:SVec2D
- Local result:Int = CalculateIntersection(line1, line2, intersection)
- If result Then
- Print "Intersection point: " + intersection.ToString()
- Else
- Print "Lines are parallel, no intersection"
- End If
- Type TLine
- Field startPoint:SVec2D
- Field endPoint:SVec2D
-
- Method New(startPoint:SVec2D, endPoint:SVec2D)
- self.startPoint = startPoint
- self.endPoint = endPoint
- End Method
- End Type
- Function CalculateIntersection:Int(line1:TLine, line2:TLine, intersection:SVec2D Var)
- Local v1:SVec2D = line1.endPoint - line1.startPoint
- Local v2:SVec2D = line2.endPoint - line2.startPoint
-
- Local crossProduct:Double = v1.x * v2.y - v1.y * v2.x
- If Abs(crossProduct) < 0.000001 Then
- Return False ' Lines are parallel
- End If
-
- Local t:Double = ((line1.startPoint.x - line2.startPoint.x) * v2.y - (line1.startPoint.y - line2.startPoint.y) * v2.x) / crossProduct
-
- intersection = line1.startPoint + v1 * t
- Return True
- End Function
- ```
- **Areas** : Computing the areas of shapes, such as triangles, polygons, and circles, often involves 2D
- vectors. For instance, the area of a triangle can be calculated using the cross product of two of its edge
- vectors, while the area of a polygon can be determined by summing the signed areas of its constituent
- triangles. Additionally, 2D vectors can be used to calculate the moments of inertia and centroids of shapes,
- which are essential properties in mechanics and engineering.
- **Dot Products** : The dot product is a fundamental operation in linear algebra that takes two vectors as
- input and returns a scalar value. In the context of 2D vectors, the dot product can be used to determine
- the angle between two vectors, project one vector onto another, or check if two vectors are perpendicular.
- Dot products are widely used in computer graphics for shading calculations and in physics simulations for
- force computations and collision responses.
- **Matrix Multiplications** : Matrix multiplication is another essential linear algebra operation that
- involves 2D vectors when dealing with transformations in a two-dimensional space. By multiplying a
- transformation matrix, such as a translation, rotation, or scaling matrix, with a position vector,
- developers can apply the corresponding transformation to a point, line, or shape in the coordinate
- system. Matrix multiplications are fundamental in computer graphics for rendering and animating scenes
- and in physics simulations for updating the positions and orientations of objects.
- ## 3D Vectors
- 3D vectors, represented by the #SVec3D, #SVec3F, and #SVec3I structs, have three components:
- x, y, and z, corresponding to coordinates in 3D space. They are essential in various applications,
- including 3D games, computer graphics, physics simulations, and engineering analyses. While some
- of the uses of 3D vectors overlap with those of 2D vectors, they extend to more complex scenarios
- and additional functionality.
- **3D Games and Animation** : In 3D games, vectors are crucial for defining the positions, velocities,
- accelerations, and orientations of game objects, such as characters, vehicles, and cameras.
- They also play a vital role in game mechanics like collision detection, pathfinding, and
- physics-based animations. In character animation, 3D vectors are used in skeleton-based systems
- for defining joint positions and rotations, enabling realistic and expressive movements.
- In this example, we define a `TGameObject` type with a position and velocity. The `Update` method is
- used to update the game object's position based on its velocity and a given time step (`dt`).
- It demonstrates moving the game object along a simple path using positions and velocities. The
- game object moves towards each point in the pathPoints array and switches
- to the next target when it gets close enough.
- ```blitzmax
- SuperStrict
- Framework BRL.StandardIO
- Import BRL.Vector
- Global pathPoints:SVec3D[] = [New SVec3D(0, 0, 0), New SVec3D(5, 10, 0), New SVec3D(10, 5, 0)]
- Global currentTargetIndex:Int = 0
- Local gameObject:TGameObject = New TGameObject(pathPoints[currentTargetIndex], New SVec3D(0, 0, 0))
- Local dt:Double = 0.5
- For Local i:Int = 1 To 30
- Local target:SVec3D = pathPoints[currentTargetIndex]
- Local direction:SVec3D = (target - gameObject.position).Normal()
-
- gameObject.velocity = direction * 1.5 ' Set the speed to 1.5 units per step
- gameObject.Update(dt)
-
- Print "Step: " + i + " | " + gameObject.ToString()
-
- If gameObject.position.DistanceTo(target) < 0.1 Then ' If the game object is close to the target, switch to the next target
- currentTargetIndex = (currentTargetIndex + 1) Mod pathPoints.Length
- End If
-
- Delay(200) ' Add a small delay for readability
- Next
- Type TGameObject
- Field position:SVec3D
- Field velocity:SVec3D
-
- Method New(position:SVec3D, velocity:SVec3D)
- self.position = position
- self.velocity = velocity
- End Method
-
- Method Update(dt:Double)
- self.position = self.position + self.velocity * dt
- End Method
-
- Method ToString:String()
- Return "Position: " + position.ToString() + " | Velocity: " + velocity.ToString()
- End Method
- End Type
- ```
- **Computer Graphics** : 3D vectors are the backbone of computer graphics, representing points,
- lines, and polygons in 3D space. They facilitate transformations, such as translation, rotation,
- and scaling, which are vital for rendering 3D models and scenes. In addition, 3D vectors are used
- in lighting calculations and surface shading, enabling the creation of realistic materials and textures.
- **Physics Simulations** : In physics simulations, 3D vectors are used to represent quantities like
- force, torque, and momentum. They enable calculations of physical properties, such as the center of
- mass and moments of inertia for rigid bodies. Moreover, 3D vectors are used in numerical methods like
- the Verlet integration and the Runge-Kutta method for simulating the motion of objects under various
- forces and constraints.
- This example demonstrates a simple Verlet integration for simulating the motion of a falling particle
- under gravity. The `TParticle` type contains position, previous position, and acceleration.
- The `Integrate` method updates the particle's position using Verlet integration. We create a particle
- and simulate its motion for 100 iterations, printing its position at each step.
- ```blitzmax
- SuperStrict
- Framework brl.standardio
- Import BRL.Vector
- Const deltaTime:Float = 0.01
- Const gravity:Float = -9.8
- Const numIterations:Int = 100
- Local p:TParticle = New TParticle(New SVec3F(0, 10, 0))
- For Local i:Int = 0 To numIterations - 1
- p.Integrate()
- Print "Iteration: " + (i + 1) + " Position: " + p.position.ToString()
- Next
- Type TParticle
- Field position:SVec3F
- Field previousPosition:SVec3F
- Field acceleration:SVec3F
-
- Method New(pos:SVec3F)
- position = pos
- previousPosition = pos
- acceleration = New SVec3F(0, gravity, 0)
- End Method
-
- Method Integrate()
- Local temp:SVec3F = position
- position = position * 2 - previousPosition + acceleration * deltaTime * deltaTime
- previousPosition = temp
- End Method
- End Type
- ```
- **Engineering Analyses** : 3D vectors play a significant role in engineering analyses, including structural,
- fluid dynamics, and thermal analyses. They are used to represent stress and strain tensors in structural
- mechanics, velocity and pressure fields in fluid dynamics, and temperature gradients in thermal analyses.
- Additionally, 3D vectors facilitate the visualization and interpretation of complex engineering data,
- like stress distributions and flow patterns.
- **Geospatial Applications** : In geospatial applications, 3D vectors are used to represent positions and
- displacements on Earth's surface and in its atmosphere. They facilitate calculations of distances, angles,
- and areas on the curved surface, as well as transformations between various coordinate systems, such as
- geographic, Cartesian, and spherical coordinates.
- **Cross Products** : Unlike 2D vectors, 3D vectors have a unique operation called the cross product.
- The cross product takes two input vectors and returns a new vector that is perpendicular to both input
- vectors and has a magnitude proportional to the sine of the angle between them. Cross products are used
- in numerous applications, including calculating surface normals, generating coordinate systems, and
- determining the torque produced by a force.
- 3D vectors are essential in various applications, extending the functionality of 2D vectors to more complex
- scenarios and additional operations.
- ## 4D Vectors
- 4D vectors, represented by the #SVec4D, #SVec4F, and #SVec4I structs, have four components: x, y, z,
- and w. While they can indeed represent points in 4D space, their primary use is for homogeneous
- coordinates in 3D graphics, along with other specialized applications.
- **Homogeneous Coordinates** : In 3D graphics, 4D vectors are employed to represent homogeneous coordinates,
- which enable affine transformations, such as translation, rotation, scaling, and perspective projection.
- By adding the w component to a 3D point, 4D vectors allow for these transformations to be represented
- by matrix multiplications, simplifying the overall process. Once the transformations are applied, the
- resulting 4D vector can be converted back to a 3D point by dividing the x, y, and z components by
- the w component.
- In this example, we define a `TTransformableObject` type with a 4D vector position to represent
- the object's position in homogeneous coordinates. The `ApplyTransformation` method applies a given
- transformation matrix to the object's position using the `Apply` method of #SMat4D. The `Get3DPosition`
- method converts the 4D vector back to a 3D point by dividing the x, y, and z components by the w component.
- We create a TransformableObject and use the #SMat4D struct functions `Translation`,
- `Rotation`, and `Scaling` to create translation, rotation, and scaling matrices. We then apply
- these transformations to the object and print its position after each transformation.
- ```blitzmax
- SuperStrict
- Framework BRL.StandardIO
- Import BRL.Vector
- Import BRL.Matrix
- Local obj:TTransformableObject = New TTransformableObject(New SVec3D(2.0, 3.0, 4.0))
- Local translationMatrix:SMat4D = SMat4D.Translation(New SVec3D(5.0, -2.0, 3.0))
- Local rotationMatrix:SMat4D = SMat4D.Rotation(New SVec3D(0.0, 1.0, 0.0), 45.0)
- Local scalingMatrix:SMat4D = SMat4D.Scaling(New SVec3D(2.0, 0.5, 1.5))
- Print "Original Position: " + obj.ToString()
- obj.ApplyTransformation(translationMatrix)
- Print "After Translation: " + obj.ToString()
- obj.ApplyTransformation(rotationMatrix)
- Print "After Rotation: " + obj.ToString()
- obj.ApplyTransformation(scalingMatrix)
- Print "After Scaling: " + obj.ToString()
- Type TTransformableObject
- Field position:SVec4D
-
- Method New(position:SVec3D)
- self.position = New SVec4D(position.x, position.y, position.z, 1.0)
- End Method
-
- Method ApplyTransformation(transformationMatrix:SMat4D)
- self.position = transformationMatrix.Apply(self.position)
- End Method
-
- Method Get3DPosition:SVec3D()
- Return New SVec3D(self.position.x / self.position.w, self.position.y / self.position.w, self.position.z / self.position.w)
- End Method
-
- Method ToString:String()
- Return "Position: " + Get3DPosition().ToString()
- End Method
- End Type
- ```
- **Perspective Projection** : Perspective projection is a crucial aspect of 3D rendering, as it creates
- the illusion of depth on a 2D screen. 4D vectors facilitate this process by allowing the perspective
- divide, which is a transformation that scales x, y, and z coordinates by the reciprocal of the w component.
- This division produces the desired depth effect, with objects appearing smaller as they move further away
- from the viewer.
- **Quaternions** : 4D vectors can also represent quaternions, which are mathematical constructs used for
- efficient 3D rotation calculations. In this context, the x, y, and z components correspond to the
- imaginary part of the quaternion, and the w component corresponds to the real part. Quaternions are
- particularly useful in computer graphics and robotics for avoiding gimbal lock, interpolating between
- orientations, and maintaining numerical stability.
- **Color Representation** : In computer graphics, 4D vectors can be used to represent colors with an
- additional alpha channel for transparency. The x, y, z, and w components correspond to the red, green,
- blue, and alpha values, respectively. This representation allows for color blending and modulation
- operations to be performed using vector arithmetic.
- **Splines and Curves** : 4D vectors can be employed to represent control points for splines and curves
- in higher-dimensional spaces. For example, in computer-aided design (CAD) and computer graphics,
- 4D vectors can define control points for NURBS (Non-uniform rational B-spline) surfaces or Bézier
- patches, enabling smooth and precise modeling of complex shapes.
- **Hyperplanes** : In machine learning and computational geometry, 4D vectors can be used to represent
- hyperplanes in 4D space. These hyperplanes can be employed for tasks like data classification,
- clustering, or nearest-neighbor searches in four-dimensional data sets.
- 4D vectors have specialized uses, primarily in homogeneous coordinates for 3D graphics, along with
- other applications such as perspective projection, quaternion representation, color representation,
- splines and curves, and hyperplanes. While sharing some similarities with 2D and 3D vectors,
- 4D vectors extend their functionality to accommodate unique scenarios and mathematical constructs.
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