From 4a8b06746f5c445e07e1565fdef414d05e99f55b Mon Sep 17 00:00:00 2001 From: Anten Skrabec Date: Tue, 22 Jun 2021 14:45:22 -0600 Subject: [PATCH] actually add the files -.- --- circle.go | 334 +++++++++++++++++++++++++++++++++++++ math.go | 15 ++ matrix.go | 98 +++++++++++ rectangle.go | 284 ++++++++++++++++++++++++++++++++ vector.go | 457 +++++++++++++++++++++++++++++++++++++++++++++++++++ 5 files changed, 1188 insertions(+) create mode 100644 circle.go create mode 100644 math.go create mode 100644 matrix.go create mode 100644 rectangle.go create mode 100644 vector.go diff --git a/circle.go b/circle.go new file mode 100644 index 0000000..8e29207 --- /dev/null +++ b/circle.go @@ -0,0 +1,334 @@ +package pixel + +import ( + "fmt" + "math" +) + +// Circle is a 2D circle. It is defined by two properties: +// - Center vector +// - Radius float64 +type Circle struct { + Center Vec + Radius float64 +} + +// C returns a new Circle with the given radius and center coordinates. +// +// Note that a negative radius is valid. +func C(center Vec, radius float64) Circle { + return Circle{ + Center: center, + Radius: radius, + } +} + +// String returns the string representation of the Circle. +// +// c := pixel.C(10.1234, pixel.ZV) +// c.String() // returns "Circle(10.12, Vec(0, 0))" +// fmt.Println(c) // Circle(10.12, Vec(0, 0)) +func (c Circle) String() string { + return fmt.Sprintf("Circle(%s, %.2f)", c.Center, c.Radius) +} + +// Norm returns the Circle in normalized form - this sets the radius to its absolute value. +// +// c := pixel.C(-10, pixel.ZV) +// c.Norm() // returns pixel.Circle{pixel.Vec{0, 0}, 10} +func (c Circle) Norm() Circle { + return Circle{ + Center: c.Center, + Radius: math.Abs(c.Radius), + } +} + +// Area returns the area of the Circle. +func (c Circle) Area() float64 { + return math.Pi * math.Pow(c.Radius, 2) +} + +// Moved returns the Circle moved by the given vector delta. +func (c Circle) Moved(delta Vec) Circle { + return Circle{ + Center: c.Center.Add(delta), + Radius: c.Radius, + } +} + +// Resized returns the Circle resized by the given delta. The Circles center is use as the anchor. +// +// c := pixel.C(pixel.ZV, 10) +// c.Resized(-5) // returns pixel.Circle{pixel.Vec{0, 0}, 5} +// c.Resized(25) // returns pixel.Circle{pixel.Vec{0, 0}, 35} +func (c Circle) Resized(radiusDelta float64) Circle { + return Circle{ + Center: c.Center, + Radius: c.Radius + radiusDelta, + } +} + +// Contains checks whether a vector `u` is contained within this Circle (including it's perimeter). +func (c Circle) Contains(u Vec) bool { + toCenter := c.Center.To(u) + return c.Radius >= toCenter.Len() +} + +// Formula returns the values of h and k, for the equation of the circle: (x-h)^2 + (y-k)^2 = r^2 +// where r is the radius of the circle. +func (c Circle) Formula() (h, k float64) { + return c.Center.X, c.Center.Y +} + +// maxCircle will return the larger circle based on the radius. +func maxCircle(c, d Circle) Circle { + if c.Radius < d.Radius { + return d + } + return c +} + +// minCircle will return the smaller circle based on the radius. +func minCircle(c, d Circle) Circle { + if c.Radius < d.Radius { + return c + } + return d +} + +// Union returns the minimal Circle which covers both `c` and `d`. +func (c Circle) Union(d Circle) Circle { + biggerC := maxCircle(c.Norm(), d.Norm()) + smallerC := minCircle(c.Norm(), d.Norm()) + + // Get distance between centers + dist := c.Center.To(d.Center).Len() + + // If the bigger Circle encompasses the smaller one, we have the result + if dist+smallerC.Radius <= biggerC.Radius { + return biggerC + } + + // Calculate radius for encompassing Circle + r := (dist + biggerC.Radius + smallerC.Radius) / 2 + + // Calculate center for encompassing Circle + theta := .5 + (biggerC.Radius-smallerC.Radius)/(2*dist) + center := Lerp(smallerC.Center, biggerC.Center, theta) + + return Circle{ + Center: center, + Radius: r, + } +} + +// Intersect returns the maximal Circle which is covered by both `c` and `d`. +// +// If `c` and `d` don't overlap, this function returns a zero-sized circle at the centerpoint between the two Circle's +// centers. +func (c Circle) Intersect(d Circle) Circle { + // Check if one of the circles encompasses the other; if so, return that one + biggerC := maxCircle(c.Norm(), d.Norm()) + smallerC := minCircle(c.Norm(), d.Norm()) + + if biggerC.Radius >= biggerC.Center.To(smallerC.Center).Len()+smallerC.Radius { + return biggerC + } + + // Calculate the midpoint between the two radii + // Distance between centers + dist := c.Center.To(d.Center).Len() + // Difference between radii + diff := dist - (c.Radius + d.Radius) + // Distance from c.Center to the weighted midpoint + distToMidpoint := c.Radius + 0.5*diff + // Weighted midpoint + center := Lerp(c.Center, d.Center, distToMidpoint/dist) + + // No need to calculate radius if the circles do not overlap + if c.Center.To(d.Center).Len() >= c.Radius+d.Radius { + return C(center, 0) + } + + radius := c.Center.To(d.Center).Len() - (c.Radius + d.Radius) + + return Circle{ + Center: center, + Radius: math.Abs(radius), + } +} + +// IntersectLine will return the shortest Vec such that if the Circle is moved by the Vec returned, the Line and Rect no +// longer intersect. +func (c Circle) IntersectLine(l Line) Vec { + return l.IntersectCircle(c).Scaled(-1) +} + +// IntersectRect returns a minimal required Vector, such that moving the circle by that vector would stop the Circle +// and the Rect intersecting. This function returns a zero-vector if the Circle and Rect do not overlap, and if only +// the perimeters touch. +// +// This function will return a non-zero vector if: +// - The Rect contains the Circle, partially or fully +// - The Circle contains the Rect, partially of fully +func (c Circle) IntersectRect(r Rect) Vec { + // Checks if the c.Center is not in the diagonal quadrants of the rectangle + if (r.Min.X <= c.Center.X && c.Center.X <= r.Max.X) || (r.Min.Y <= c.Center.Y && c.Center.Y <= r.Max.Y) { + // 'grow' the Rect by c.Radius in each orthagonal + grown := Rect{Min: r.Min.Sub(V(c.Radius, c.Radius)), Max: r.Max.Add(V(c.Radius, c.Radius))} + if !grown.Contains(c.Center) { + // c.Center not close enough to overlap, return zero-vector + return ZV + } + + // Get minimum distance to travel out of Rect + rToC := r.Center().To(c.Center) + h := c.Radius - math.Abs(rToC.X) + (r.W() / 2) + v := c.Radius - math.Abs(rToC.Y) + (r.H() / 2) + + if rToC.X < 0 { + h = -h + } + if rToC.Y < 0 { + v = -v + } + + // No intersect + if h == 0 && v == 0 { + return ZV + } + + if math.Abs(h) > math.Abs(v) { + // Vertical distance shorter + return V(0, v) + } + return V(h, 0) + } else { + // The center is in the diagonal quadrants + + // Helper points to make code below easy to read. + rectTopLeft := V(r.Min.X, r.Max.Y) + rectBottomRight := V(r.Max.X, r.Min.Y) + + // Check for overlap. + if !(c.Contains(r.Min) || c.Contains(r.Max) || c.Contains(rectTopLeft) || c.Contains(rectBottomRight)) { + // No overlap. + return ZV + } + + var centerToCorner Vec + if c.Center.To(r.Min).Len() <= c.Radius { + // Closest to bottom-left + centerToCorner = c.Center.To(r.Min) + } + if c.Center.To(r.Max).Len() <= c.Radius { + // Closest to top-right + centerToCorner = c.Center.To(r.Max) + } + if c.Center.To(rectTopLeft).Len() <= c.Radius { + // Closest to top-left + centerToCorner = c.Center.To(rectTopLeft) + } + if c.Center.To(rectBottomRight).Len() <= c.Radius { + // Closest to bottom-right + centerToCorner = c.Center.To(rectBottomRight) + } + + cornerToCircumferenceLen := c.Radius - centerToCorner.Len() + + return centerToCorner.Unit().Scaled(cornerToCircumferenceLen) + } +} + +// IntersectionPoints returns all the points where the Circle intersects with the line provided. This can be zero, one or +// two points, depending on the location of the shapes. The points of intersection will be returned in order of +// closest-to-l.A to closest-to-l.B. +func (c Circle) IntersectionPoints(l Line) []Vec { + cContainsA := c.Contains(l.A) + cContainsB := c.Contains(l.B) + + // Special case for both endpoint being contained within the circle + if cContainsA && cContainsB { + return []Vec{} + } + + // Get closest point on the line to this circles' center + closestToCenter := l.Closest(c.Center) + + // If the distance to the closest point is greater than the radius, there are no points of intersection + if closestToCenter.To(c.Center).Len() > c.Radius { + return []Vec{} + } + + // If the distance to the closest point is equal to the radius, the line is tangent and the closest point is the + // point at which it touches the circle. + if closestToCenter.To(c.Center).Len() == c.Radius { + return []Vec{closestToCenter} + } + + // Special case for endpoint being on the circles' center + if c.Center == l.A || c.Center == l.B { + otherEnd := l.B + if c.Center == l.B { + otherEnd = l.A + } + intersect := c.Center.Add(c.Center.To(otherEnd).Unit().Scaled(c.Radius)) + return []Vec{intersect} + } + + // This means the distance to the closest point is less than the radius, so there is at least one intersection, + // possibly two. + + // If one of the end points exists within the circle, there is only one intersection + if cContainsA || cContainsB { + containedPoint := l.A + otherEnd := l.B + if cContainsB { + containedPoint = l.B + otherEnd = l.A + } + + // Use trigonometry to get the length of the line between the contained point and the intersection point. + // The following is used to describe the triangle formed: + // - a is the side between contained point and circle center + // - b is the side between the center and the intersection point (radius) + // - c is the side between the contained point and the intersection point + // The captials of these letters are used as the angles opposite the respective sides. + // a and b are known + a := containedPoint.To(c.Center).Len() + b := c.Radius + // B can be calculated by subtracting the angle of b (to the x-axis) from the angle of c (to the x-axis) + B := containedPoint.To(c.Center).Angle() - containedPoint.To(otherEnd).Angle() + // Using the Sin rule we can get A + A := math.Asin((a * math.Sin(B)) / b) + // Using the rule that there are 180 degrees (or Pi radians) in a triangle, we can now get C + C := math.Pi - A + B + // If C is zero, the line segment is in-line with the center-intersect line. + var c float64 + if C == 0 { + c = b - a + } else { + // Using the Sine rule again, we can now get c + c = (a * math.Sin(C)) / math.Sin(A) + } + // Travelling from the contained point to the other end by length of a will provide the intersection point. + return []Vec{ + containedPoint.Add(containedPoint.To(otherEnd).Unit().Scaled(c)), + } + } + + // Otherwise the endpoints exist outside of the circle, and the line segment intersects in two locations. + // The vector formed by going from the closest point to the center of the circle will be perpendicular to the line; + // this forms a right-angled triangle with the intersection points, with the radius as the hypotenuse. + // Calculate the other triangles' sides' length. + a := math.Sqrt(math.Pow(c.Radius, 2) - math.Pow(closestToCenter.To(c.Center).Len(), 2)) + + // Travelling in both directions from the closest point by length of a will provide the two intersection points. + first := closestToCenter.Add(closestToCenter.To(l.A).Unit().Scaled(a)) + second := closestToCenter.Add(closestToCenter.To(l.B).Unit().Scaled(a)) + + if first.To(l.A).Len() < second.To(l.A).Len() { + return []Vec{first, second} + } + return []Vec{second, first} +} diff --git a/math.go b/math.go new file mode 100644 index 0000000..69784f3 --- /dev/null +++ b/math.go @@ -0,0 +1,15 @@ +package pixel + +// Clamp returns x clamped to the interval [min, max]. +// +// If x is less than min, min is returned. If x is more than max, max is returned. Otherwise, x is +// returned. +func Clamp(x, min, max float64) float64 { + if x < min { + return min + } + if x > max { + return max + } + return x +} diff --git a/matrix.go b/matrix.go new file mode 100644 index 0000000..3f02fea --- /dev/null +++ b/matrix.go @@ -0,0 +1,98 @@ +package pixel + +import ( + "fmt" + "math" +) + +// Matrix is a 2x3 affine matrix that can be used for all kinds of spatial transforms, such +// as movement, scaling and rotations. +// +// Matrix has a handful of useful methods, each of which adds a transformation to the matrix. For +// example: +// +// pixel.IM.Moved(pixel.V(100, 200)).Rotated(pixel.ZV, math.Pi/2) +// +// This code creates a Matrix that first moves everything by 100 units horizontally and 200 units +// vertically and then rotates everything by 90 degrees around the origin. +// +// Layout is: +// [0] [2] [4] +// [1] [3] [5] +// 0 0 1 (implicit row) +type Matrix [6]float64 + +// IM stands for identity matrix. Does nothing, no transformation. +var IM = Matrix{1, 0, 0, 1, 0, 0} + +// String returns a string representation of the Matrix. +// +// m := pixel.IM +// fmt.Println(m) // Matrix(1 0 0 | 0 1 0) +func (m Matrix) String() string { + return fmt.Sprintf( + "Matrix(%v %v %v | %v %v %v)", + m[0], m[2], m[4], + m[1], m[3], m[5], + ) +} + +// Moved moves everything by the delta vector. +func (m Matrix) Moved(delta Vec) Matrix { + m[4], m[5] = m[4]+delta.X, m[5]+delta.Y + return m +} + +// ScaledXY scales everything around a given point by the scale factor in each axis respectively. +func (m Matrix) ScaledXY(around Vec, scale Vec) Matrix { + m[4], m[5] = m[4]-around.X, m[5]-around.Y + m[0], m[2], m[4] = m[0]*scale.X, m[2]*scale.X, m[4]*scale.X + m[1], m[3], m[5] = m[1]*scale.Y, m[3]*scale.Y, m[5]*scale.Y + m[4], m[5] = m[4]+around.X, m[5]+around.Y + return m +} + +// Scaled scales everything around a given point by the scale factor. +func (m Matrix) Scaled(around Vec, scale float64) Matrix { + return m.ScaledXY(around, V(scale, scale)) +} + +// Rotated rotates everything around a given point by the given angle in radians. +func (m Matrix) Rotated(around Vec, angle float64) Matrix { + sint, cost := math.Sincos(angle) + m[4], m[5] = m[4]-around.X, m[5]-around.Y + m = m.Chained(Matrix{cost, sint, -sint, cost, 0, 0}) + m[4], m[5] = m[4]+around.X, m[5]+around.Y + return m +} + +// Chained adds another Matrix to this one. All tranformations by the next Matrix will be applied +// after the transformations of this Matrix. +func (m Matrix) Chained(next Matrix) Matrix { + return Matrix{ + next[0]*m[0] + next[2]*m[1], + next[1]*m[0] + next[3]*m[1], + next[0]*m[2] + next[2]*m[3], + next[1]*m[2] + next[3]*m[3], + next[0]*m[4] + next[2]*m[5] + next[4], + next[1]*m[4] + next[3]*m[5] + next[5], + } +} + +// Project applies all transformations added to the Matrix to a vector u and returns the result. +// +// Time complexity is O(1). +func (m Matrix) Project(u Vec) Vec { + return Vec{m[0]*u.X + m[2]*u.Y + m[4], m[1]*u.X + m[3]*u.Y + m[5]} +} + +// Unproject does the inverse operation to Project. +// +// Time complexity is O(1). +func (m Matrix) Unproject(u Vec) Vec { + det := m[0]*m[3] - m[2]*m[1] + return Vec{ + (m[3]*(u.X-m[4]) - m[2]*(u.Y-m[5])) / det, + (-m[1]*(u.X-m[4]) + m[0]*(u.Y-m[5])) / det, + } +} diff --git a/rectangle.go b/rectangle.go new file mode 100644 index 0000000..b76acb9 --- /dev/null +++ b/rectangle.go @@ -0,0 +1,284 @@ +package pixel + +import ( + "fmt" + "math" +) + +// Rect is a 2D rectangle aligned with the axes of the coordinate system. It is defined by two +// points, Min and Max. +// +// The invariant should hold, that Max's components are greater or equal than Min's components +// respectively. +type Rect struct { + Min, Max Vec +} + +// ZR is a zero rectangle. +var ZR = Rect{Min: ZV, Max: ZV} + +// R returns a new Rect with given the Min and Max coordinates. +// +// Note that the returned rectangle is not automatically normalized. +func R(minX, minY, maxX, maxY float64) Rect { + return Rect{ + Min: Vec{minX, minY}, + Max: Vec{maxX, maxY}, + } +} + +// String returns the string representation of the Rect. +// +// r := pixel.R(100, 50, 200, 300) +// r.String() // returns "Rect(100, 50, 200, 300)" +// fmt.Println(r) // Rect(100, 50, 200, 300) +func (r Rect) String() string { + return fmt.Sprintf("Rect(%v, %v, %v, %v)", r.Min.X, r.Min.Y, r.Max.X, r.Max.Y) +} + +// Norm returns the Rect in normal form, such that Max is component-wise greater or equal than Min. +func (r Rect) Norm() Rect { + return Rect{ + Min: Vec{ + math.Min(r.Min.X, r.Max.X), + math.Min(r.Min.Y, r.Max.Y), + }, + Max: Vec{ + math.Max(r.Min.X, r.Max.X), + math.Max(r.Min.Y, r.Max.Y), + }, + } +} + +// W returns the width of the Rect. +func (r Rect) W() float64 { + return r.Max.X - r.Min.X +} + +// H returns the height of the Rect. +func (r Rect) H() float64 { + return r.Max.Y - r.Min.Y +} + +// Size returns the vector of width and height of the Rect. +func (r Rect) Size() Vec { + return V(r.W(), r.H()) +} + +// Area returns the area of r. If r is not normalized, area may be negative. +func (r Rect) Area() float64 { + return r.W() * r.H() +} + +// Edges will return the four lines which make up the edges of the rectangle. +func (r Rect) Edges() [4]Line { + corners := r.Vertices() + + return [4]Line{ + {A: corners[0], B: corners[1]}, + {A: corners[1], B: corners[2]}, + {A: corners[2], B: corners[3]}, + {A: corners[3], B: corners[0]}, + } +} + +// Anchor is a vector used to define anchors, such as `Center`, `Top`, `TopRight`, etc. +type Anchor Vec + +var ( + Center = Anchor{0.5, 0.5} + Top = Anchor{0.5, 0} + TopRight = Anchor{0, 0} + Right = Anchor{0, 0.5} + BottomRight = Anchor{0, 1} + Bottom = Anchor{0.5, 1} + BottomLeft = Anchor{1, 1} + Left = Anchor{1, 0.5} + TopLeft = Anchor{1, 0} +) + +var anchorStrings map[Anchor]string = map[Anchor]string{ + Center: "center", + Top: "top", + TopRight: "top-right", + Right: "right", + BottomRight: "bottom-right", + Bottom: "bottom", + BottomLeft: "bottom-left", + Left: "left", + TopLeft: "top-left", +} + +// String returns the string representation of an anchor. +func (anchor Anchor) String() string { + return anchorStrings[anchor] +} + +var oppositeAnchors map[Anchor]Anchor = map[Anchor]Anchor{ + Center: Center, + Top: Bottom, + Bottom: Top, + Right: Left, + Left: Right, + TopRight: BottomLeft, + BottomLeft: TopRight, + BottomRight: TopLeft, + TopLeft: BottomRight, +} + +// Opposite returns the opposite position of the anchor (ie. Top -> Bottom; BottomLeft -> TopRight, etc.). +func (anchor Anchor) Opposite() Anchor { + return oppositeAnchors[anchor] +} + +// AnchorPos returns the relative position of the given anchor. +func (r Rect) AnchorPos(anchor Anchor) Vec { + return r.Size().ScaledXY(V(0, 0).Sub(Vec(anchor))) +} + +// AlignedTo returns the rect moved by the given anchor. +func (rect Rect) AlignedTo(anchor Anchor) Rect { + return rect.Moved(rect.AnchorPos(anchor)) +} + +// Center returns the position of the center of the Rect. +// `rect.Center()` is equivalent to `rect.Anchor(pixel.Anchor.Center)` +func (r Rect) Center() Vec { + return Lerp(r.Min, r.Max, 0.5) +} + +// Moved returns the Rect moved (both Min and Max) by the given vector delta. +func (r Rect) Moved(delta Vec) Rect { + return Rect{ + Min: r.Min.Add(delta), + Max: r.Max.Add(delta), + } +} + +// Resized returns the Rect resized to the given size while keeping the position of the given +// anchor. +// +// r.Resized(r.Min, size) // resizes while keeping the position of the lower-left corner +// r.Resized(r.Max, size) // same with the top-right corner +// r.Resized(r.Center(), size) // resizes around the center +// +// This function does not make sense for resizing a rectangle of zero area and will panic. Use +// ResizedMin in the case of zero area. +func (r Rect) Resized(anchor, size Vec) Rect { + if r.W()*r.H() == 0 { + panic(fmt.Errorf("(%T).Resize: zero area", r)) + } + fraction := Vec{size.X / r.W(), size.Y / r.H()} + return Rect{ + Min: anchor.Add(r.Min.Sub(anchor).ScaledXY(fraction)), + Max: anchor.Add(r.Max.Sub(anchor).ScaledXY(fraction)), + } +} + +// ResizedMin returns the Rect resized to the given size while keeping the position of the Rect's +// Min. +// +// Sizes of zero area are safe here. +func (r Rect) ResizedMin(size Vec) Rect { + return Rect{ + Min: r.Min, + Max: r.Min.Add(size), + } +} + +// Contains checks whether a vector u is contained within this Rect (including it's borders). +func (r Rect) Contains(u Vec) bool { + return r.Min.X <= u.X && u.X <= r.Max.X && r.Min.Y <= u.Y && u.Y <= r.Max.Y +} + +// Union returns the minimal Rect which covers both r and s. Rects r and s must be normalized. +func (r Rect) Union(s Rect) Rect { + return R( + math.Min(r.Min.X, s.Min.X), + math.Min(r.Min.Y, s.Min.Y), + math.Max(r.Max.X, s.Max.X), + math.Max(r.Max.Y, s.Max.Y), + ) +} + +// Intersect returns the maximal Rect which is covered by both r and s. Rects r and s must be normalized. +// +// If r and s don't overlap, this function returns a zero-rectangle. +func (r Rect) Intersect(s Rect) Rect { + t := R( + math.Max(r.Min.X, s.Min.X), + math.Max(r.Min.Y, s.Min.Y), + math.Min(r.Max.X, s.Max.X), + math.Min(r.Max.Y, s.Max.Y), + ) + if t.Min.X >= t.Max.X || t.Min.Y >= t.Max.Y { + return ZR + } + return t +} + +// Intersects returns whether or not the given Rect intersects at any point with this Rect. +// +// This function is overall about 5x faster than Intersect, so it is better +// to use if you have no need for the returned Rect from Intersect. +func (r Rect) Intersects(s Rect) bool { + return !(s.Max.X < r.Min.X || + s.Min.X > r.Max.X || + s.Max.Y < r.Min.Y || + s.Min.Y > r.Max.Y) +} + +// IntersectCircle returns a minimal required Vector, such that moving the rect by that vector would stop the Circle +// and the Rect intersecting. This function returns a zero-vector if the Circle and Rect do not overlap, and if only +// the perimeters touch. +// +// This function will return a non-zero vector if: +// - The Rect contains the Circle, partially or fully +// - The Circle contains the Rect, partially of fully +func (r Rect) IntersectCircle(c Circle) Vec { + return c.IntersectRect(r).Scaled(-1) +} + +// IntersectLine will return the shortest Vec such that if the Rect is moved by the Vec returned, the Line and Rect no +// longer intersect. +func (r Rect) IntersectLine(l Line) Vec { + return l.IntersectRect(r).Scaled(-1) +} + +// IntersectionPoints returns all the points where the Rect intersects with the line provided. This can be zero, one or +// two points, depending on the location of the shapes. The points of intersection will be returned in order of +// closest-to-l.A to closest-to-l.B. +func (r Rect) IntersectionPoints(l Line) []Vec { + // Use map keys to ensure unique points + pointMap := make(map[Vec]struct{}) + + for _, edge := range r.Edges() { + if intersect, ok := l.Intersect(edge); ok { + pointMap[intersect] = struct{}{} + } + } + + points := make([]Vec, 0, len(pointMap)) + for point := range pointMap { + points = append(points, point) + } + + // Order the points + if len(points) == 2 { + if points[1].To(l.A).Len() < points[0].To(l.A).Len() { + return []Vec{points[1], points[0]} + } + } + + return points +} + +// Vertices returns a slice of the four corners which make up the rectangle. +func (r Rect) Vertices() [4]Vec { + return [4]Vec{ + r.Min, + V(r.Min.X, r.Max.Y), + r.Max, + V(r.Max.X, r.Min.Y), + } +} diff --git a/vector.go b/vector.go new file mode 100644 index 0000000..d9081b3 --- /dev/null +++ b/vector.go @@ -0,0 +1,457 @@ +package pixel + +import ( + "fmt" + "math" +) + +// Vec is a 2D vector type with X and Y coordinates. +// +// Create vectors with the V constructor: +// +// u := pixel.V(1, 2) +// v := pixel.V(8, -3) +// +// Use various methods to manipulate them: +// +// w := u.Add(v) +// fmt.Println(w) // Vec(9, -1) +// fmt.Println(u.Sub(v)) // Vec(-7, 5) +// u = pixel.V(2, 3) +// v = pixel.V(8, 1) +// if u.X < 0 { +// fmt.Println("this won't happen") +// } +// x := u.Unit().Dot(v.Unit()) +type Vec struct { + X, Y float64 +} + +// ZV is a zero vector. +var ZV = Vec{0, 0} + +// V returns a new 2D vector with the given coordinates. +func V(x, y float64) Vec { + return Vec{x, y} +} + +// nearlyEqual compares two float64s and returns whether they are equal, accounting for rounding errors.At worst, the +// result is correct to 7 significant digits. +func nearlyEqual(a, b float64) bool { + epsilon := 0.000001 + + if a == b { + return true + } + + diff := math.Abs(a - b) + + if a == 0.0 || b == 0.0 || diff < math.SmallestNonzeroFloat64 { + return diff < (epsilon * math.SmallestNonzeroFloat64) + } + + absA := math.Abs(a) + absB := math.Abs(b) + + return diff/math.Min(absA+absB, math.MaxFloat64) < epsilon +} + +// Eq will compare two vectors and return whether they are equal accounting for rounding errors. At worst, the result +// is correct to 7 significant digits. +func (u Vec) Eq(v Vec) bool { + return nearlyEqual(u.X, v.X) && nearlyEqual(u.Y, v.Y) +} + +// Unit returns a vector of length 1 facing the given angle. +func Unit(angle float64) Vec { + return Vec{1, 0}.Rotated(angle) +} + +// String returns the string representation of the vector u. +// +// u := pixel.V(4.5, -1.3) +// u.String() // returns "Vec(4.5, -1.3)" +// fmt.Println(u) // Vec(4.5, -1.3) +func (u Vec) String() string { + return fmt.Sprintf("Vec(%v, %v)", u.X, u.Y) +} + +// XY returns the components of the vector in two return values. +func (u Vec) XY() (x, y float64) { + return u.X, u.Y +} + +// Add returns the sum of vectors u and v. +func (u Vec) Add(v Vec) Vec { + return Vec{ + u.X + v.X, + u.Y + v.Y, + } +} + +// Sub returns the difference betweeen vectors u and v. +func (u Vec) Sub(v Vec) Vec { + return Vec{ + u.X - v.X, + u.Y - v.Y, + } +} + +// Floor converts x and y to their integer equivalents. +func (u Vec) Floor() Vec { + return Vec{ + math.Floor(u.X), + math.Floor(u.Y), + } +} + +// To returns the vector from u to v. Equivalent to v.Sub(u). +func (u Vec) To(v Vec) Vec { + return Vec{ + v.X - u.X, + v.Y - u.Y, + } +} + +// Scaled returns the vector u multiplied by c. +func (u Vec) Scaled(c float64) Vec { + return Vec{u.X * c, u.Y * c} +} + +// ScaledXY returns the vector u multiplied by the vector v component-wise. +func (u Vec) ScaledXY(v Vec) Vec { + return Vec{u.X * v.X, u.Y * v.Y} +} + +// Len returns the length of the vector u. +func (u Vec) Len() float64 { + return math.Hypot(u.X, u.Y) +} + +// Angle returns the angle between the vector u and the x-axis. The result is in range [-Pi, Pi]. +func (u Vec) Angle() float64 { + return math.Atan2(u.Y, u.X) +} + +// Unit returns a vector of length 1 facing the direction of u (has the same angle). +func (u Vec) Unit() Vec { + if u.X == 0 && u.Y == 0 { + return Vec{1, 0} + } + return u.Scaled(1 / u.Len()) +} + +// Rotated returns the vector u rotated by the given angle in radians. +func (u Vec) Rotated(angle float64) Vec { + sin, cos := math.Sincos(angle) + return Vec{ + u.X*cos - u.Y*sin, + u.X*sin + u.Y*cos, + } +} + +// Normal returns a vector normal to u. Equivalent to u.Rotated(math.Pi / 2), but faster. +func (u Vec) Normal() Vec { + return Vec{-u.Y, u.X} +} + +// Dot returns the dot product of vectors u and v. +func (u Vec) Dot(v Vec) float64 { + return u.X*v.X + u.Y*v.Y +} + +// Cross return the cross product of vectors u and v. +func (u Vec) Cross(v Vec) float64 { + return u.X*v.Y - v.X*u.Y +} + +// Project returns a projection (or component) of vector u in the direction of vector v. +// +// Behaviour is undefined if v is a zero vector. +func (u Vec) Project(v Vec) Vec { + len := u.Dot(v) / v.Len() + return v.Unit().Scaled(len) +} + +// Map applies the function f to both x and y components of the vector u and returns the modified +// vector. +// +// u := pixel.V(10.5, -1.5) +// v := u.Map(math.Floor) // v is Vec(10, -2), both components of u floored +func (u Vec) Map(f func(float64) float64) Vec { + return Vec{ + f(u.X), + f(u.Y), + } +} + +// Lerp returns a linear interpolation between vectors a and b. +// +// This function basically returns a point along the line between a and b and t chooses which one. +// If t is 0, then a will be returned, if t is 1, b will be returned. Anything between 0 and 1 will +// return the appropriate point between a and b and so on. +func Lerp(a, b Vec, t float64) Vec { + return a.Scaled(1 - t).Add(b.Scaled(t)) +} + +// Line is a 2D line segment, between points A and B. +type Line struct { + A, B Vec +} + +// L creates and returns a new Line. +func L(from, to Vec) Line { + return Line{ + A: from, + B: to, + } +} + +// Bounds returns the lines bounding box. This is in the form of a normalized Rect. +func (l Line) Bounds() Rect { + return R(l.A.X, l.A.Y, l.B.X, l.B.Y).Norm() +} + +// Center will return the point at center of the line; that is, the point equidistant from either end. +func (l Line) Center() Vec { + return l.A.Add(l.A.To(l.B).Scaled(0.5)) +} + +// Closest will return the point on the line which is closest to the Vec provided. +func (l Line) Closest(v Vec) Vec { + // between is a helper function which determines whether x is greater than min(a, b) and less than max(a, b) + between := func(a, b, x float64) bool { + min := math.Min(a, b) + max := math.Max(a, b) + return min < x && x < max + } + + // Closest point will be on a line which perpendicular to this line. + // If and only if the infinite perpendicular line intersects the segment. + m, b := l.Formula() + + // Account for horizontal lines + if m == 0 { + x := v.X + y := l.A.Y + + // check if the X coordinate of v is on the line + if between(l.A.X, l.B.X, v.X) { + return V(x, y) + } + + // Otherwise get the closest endpoint + if l.A.To(v).Len() < l.B.To(v).Len() { + return l.A + } + return l.B + } + + // Account for vertical lines + if math.IsInf(math.Abs(m), 1) { + x := l.A.X + y := v.Y + + // check if the Y coordinate of v is on the line + if between(l.A.Y, l.B.Y, v.Y) { + return V(x, y) + } + + // Otherwise get the closest endpoint + if l.A.To(v).Len() < l.B.To(v).Len() { + return l.A + } + return l.B + } + + perpendicularM := -1 / m + perpendicularB := v.Y - (perpendicularM * v.X) + + // Coordinates of intersect (of infinite lines) + x := (perpendicularB - b) / (m - perpendicularM) + y := m*x + b + + // Check if the point lies between the x and y bounds of the segment + if !between(l.A.X, l.B.X, x) && !between(l.A.Y, l.B.Y, y) { + // Not within bounding box + toStart := v.To(l.A) + toEnd := v.To(l.B) + + if toStart.Len() < toEnd.Len() { + return l.A + } + return l.B + } + + return V(x, y) +} + +// Contains returns whether the provided Vec lies on the line. +func (l Line) Contains(v Vec) bool { + return l.Closest(v).Eq(v) +} + +// Formula will return the values that represent the line in the formula: y = mx + b +// This function will return math.Inf+, math.Inf- for a vertical line. +func (l Line) Formula() (m, b float64) { + // Account for horizontal lines + if l.B.Y == l.A.Y { + return 0, l.A.Y + } + + m = (l.B.Y - l.A.Y) / (l.B.X - l.A.X) + b = l.A.Y - (m * l.A.X) + + return m, b +} + +// Intersect will return the point of intersection for the two line segments. If the line segments do not intersect, +// this function will return the zero-vector and false. +func (l Line) Intersect(k Line) (Vec, bool) { + // Check if the lines are parallel + lDir := l.A.To(l.B) + kDir := k.A.To(k.B) + if lDir.X == kDir.X && lDir.Y == kDir.Y { + return ZV, false + } + + // The lines intersect - but potentially not within the line segments. + // Get the intersection point for the lines if they were infinitely long, check if the point exists on both of the + // segments + lm, lb := l.Formula() + km, kb := k.Formula() + + // Account for vertical lines + if math.IsInf(math.Abs(lm), 1) && math.IsInf(math.Abs(km), 1) { + // Both vertical, therefore parallel + return ZV, false + } + + var x, y float64 + + if math.IsInf(math.Abs(lm), 1) || math.IsInf(math.Abs(km), 1) { + // One line is vertical + intersectM := lm + intersectB := lb + verticalLine := k + + if math.IsInf(math.Abs(lm), 1) { + intersectM = km + intersectB = kb + verticalLine = l + } + + y = intersectM*verticalLine.A.X + intersectB + x = verticalLine.A.X + } else { + // Coordinates of intersect + x = (kb - lb) / (lm - km) + y = lm*x + lb + } + + if l.Contains(V(x, y)) && k.Contains(V(x, y)) { + // The intersect point is on both line segments, they intersect. + return V(x, y), true + } + + return ZV, false +} + +// IntersectCircle will return the shortest Vec such that moving the Line by that Vec will cause the Line and Circle +// to no longer intesect. If they do not intersect at all, this function will return a zero-vector. +func (l Line) IntersectCircle(c Circle) Vec { + // Get the point on the line closest to the center of the circle. + closest := l.Closest(c.Center) + cirToClosest := c.Center.To(closest) + + if cirToClosest.Len() >= c.Radius { + return ZV + } + + return cirToClosest.Scaled(cirToClosest.Len() - c.Radius) +} + +// IntersectRect will return the shortest Vec such that moving the Line by that Vec will cause the Line and Rect to +// no longer intesect. If they do not intersect at all, this function will return a zero-vector. +func (l Line) IntersectRect(r Rect) Vec { + // Check if either end of the line segment are within the rectangle + if r.Contains(l.A) || r.Contains(l.B) { + // Use the Rect.Intersect to get minimal return value + rIntersect := l.Bounds().Intersect(r) + if rIntersect.H() > rIntersect.W() { + // Go vertical + return V(0, rIntersect.H()) + } + return V(rIntersect.W(), 0) + } + + // Check if any of the rectangles' edges intersect with this line. + for _, edge := range r.Edges() { + if _, ok := l.Intersect(edge); ok { + // Get the closest points on the line to each corner, where: + // - the point is contained by the rectangle + // - the point is not the corner itself + corners := r.Vertices() + var closest *Vec + closestCorner := corners[0] + for _, c := range corners { + cc := l.Closest(c) + if closest == nil || (closest.Len() > cc.Len() && r.Contains(cc)) { + closest = &cc + closestCorner = c + } + } + + return closest.To(closestCorner) + } + } + + // No intersect + return ZV +} + +// Len returns the length of the line segment. +func (l Line) Len() float64 { + return l.A.To(l.B).Len() +} + +// Moved will return a line moved by the delta Vec provided. +func (l Line) Moved(delta Vec) Line { + return Line{ + A: l.A.Add(delta), + B: l.B.Add(delta), + } +} + +// Rotated will rotate the line around the provided Vec. +func (l Line) Rotated(around Vec, angle float64) Line { + // Move the line so we can use `Vec.Rotated` + lineShifted := l.Moved(around.Scaled(-1)) + + lineRotated := Line{ + A: lineShifted.A.Rotated(angle), + B: lineShifted.B.Rotated(angle), + } + + return lineRotated.Moved(around) +} + +// Scaled will return the line scaled around the center point. +func (l Line) Scaled(scale float64) Line { + return l.ScaledXY(l.Center(), scale) +} + +// ScaledXY will return the line scaled around the Vec provided. +func (l Line) ScaledXY(around Vec, scale float64) Line { + toA := around.To(l.A).Scaled(scale) + toB := around.To(l.B).Scaled(scale) + + return Line{ + A: around.Add(toA), + B: around.Add(toB), + } +} + +func (l Line) String() string { + return fmt.Sprintf("Line(%v, %v)", l.A, l.B) +}