總覽
-
Bézier 曲線是一種用於計算機圖形學的參數曲線。
-
在本次作業中，你需要實現de Casteljau 算法來繪制由4 個控制點表示的Bézier 曲線(當你正確實現該算法時，你可以支持繪制由更多點來控制的Bézier 曲線)。
-
你需要修改的函數在提供的main.cpp 文件中。
bezier：該函數實現繪制Bézier 曲線的功能。
它使用一個控制點序列和一個OpenCV::Mat 對象作為輸入，沒有返回值。它會使t 在0 到1 的范圍內進行迭代，並在每次迭代中使t 增加一個微小值。對於每個需要計算的t，將調用另一個函數recursive_bezier，然后該函數將返回在Bézier 曲線上t處的點。最后，將返回的點繪制在OpenCV ：：Mat 對象上。- recursive_bezier：該函數使用一個控制點序列和一個浮點數t 作為輸入，實現de Casteljau 算法來返回Bézier 曲線上對應點的坐標。
實現
-
版本 4.26.2
-
naive_bezier
-
數學公式
-
代碼
void AActor_BezierCuve::naive_bezier() { FVector& p_0 = m_points[0]; FVector& p_1 = m_points[1]; FVector& p_2 = m_points[2]; FVector& p_3 = m_points[3]; FVector& p_4 = m_points[4]; for (double t = 0.0; t <= 1.0; t += 0.001) { auto point = std::pow(1 - t, 4) * p_0 + 4 * t * std::pow(1 - t, 3) * p_1 + 6 * std::pow(t, 2) * std::pow((1 - t), 2) * p_2 + 4 * std::pow(t, 3) * (1 - t) * p_3 + std::pow(t, 4) * p_4; DrawDebugPoint(GetWorld(), point, 2.0f, FColor::Green,true,5.0f); //UKismetSystemLibrary::PrintString(GetWorld(), point.ToString()); } }
-
-
recursive_bezier
-
De Casteljau 算法說明如下：
- 考慮一個p0, p1, ... pn 為控制點序列的Bézier 曲線。首先，將相鄰的點連接起來以形成線段。
- 用t : (1 − t) 的比例細分每個線段，並找到該分割點。
- 得到的分割點作為新的控制點序列，新序列的長度會減少一。
- 如果序列只包含一個點，則返回該點並終止。否則，使用新的控制點序列並轉到步驟1。使用[0,1] 中的多個不同的t 來執行上述算法，你就能得到相應的Bézier 曲線。
-
代碼
void AActor_BezierCuve::bezier() { for (double t = 0.0; t <= 1.0; t += 0.001) { FVector point = recursive_bezier(m_points, t); DrawDebugPoint(GetWorld(), point, 2.0f, FColor(10,214,255,255),true,5.0f); } } // De Casteljau 算法，遞歸 FVector AActor_BezierCuve::recursive_bezier(TArray<FVector>& points, float t) { if (points.Num() < 3) { return (1 - t) * points[0] + t * points[1]; } TArray<FVector> newPoint; for (int i = 0; i < points.Num() - 1; i++) { newPoint.Add((1 - t) * points[i] + t * points[i + 1]); } return recursive_bezier(newPoint, t); }
-
-
最終效果
附錄
所有代碼
-
Actor_BezierCuve.h
點擊查看代碼
```cpp UCLASS() class GAMES101_API AActor_BezierCuve : public AActor { GENERATED_BODY() public: // Sets default values for this actor's properties AActor_BezierCuve(); protected: // Called when the game starts or when spawned virtual void BeginPlay() override; public: // Called every frame virtual void Tick(float DeltaTime) override; UFUNCTION(BlueprintCallable) void naive_bezier(); UFUNCTION(BlueprintCallable) void bezier(); UFUNCTION(BlueprintCallable) FVector recursive_bezier(TArray<FVector>& points,float t); public: UPROPERTY(VisibleAnywhere) USceneComponent* root; UPROPERTY(VisibleAnywhere) UStaticMeshComponent* point0; UPROPERTY(VisibleAnywhere) UStaticMeshComponent* point1; UPROPERTY(VisibleAnywhere) UStaticMeshComponent* point2; UPROPERTY(VisibleAnywhere) UStaticMeshComponent* point3; UPROPERTY(VisibleAnywhere) UStaticMeshComponent* point4; UPROPERTY(); TArray<FVector> m_points; UPROPERTY(EditAnywhere); bool m_bUseRecursiveBezier; }; ``` -
AActor_BezierCuve.cpp
點擊查看代碼
#include "Actor_BezierCuve.h" #include "DrawDebugHelpers.h" #include <cmath> #include "Kismet/KismetSystemLibrary.h" // Sets default values AActor_BezierCuve::AActor_BezierCuve() { // Set this actor to call Tick() every frame. You can turn this off to improve performance if you don't need it. PrimaryActorTick.bCanEverTick = true; root = CreateDefaultSubobject<USceneComponent>(TEXT("root")); SetRootComponent(root); point0 = CreateDefaultSubobject<UStaticMeshComponent>(TEXT("point0")); point1 = CreateDefaultSubobject<UStaticMeshComponent>(TEXT("point1")); point2 = CreateDefaultSubobject<UStaticMeshComponent>(TEXT("point2")); point3 = CreateDefaultSubobject<UStaticMeshComponent>(TEXT("point3")); point4 = CreateDefaultSubobject<UStaticMeshComponent>(TEXT("point4")); point0->SetupAttachment(root); point1->SetupAttachment(root); point2->SetupAttachment(root); point3->SetupAttachment(root); point4->SetupAttachment(root); m_points.Init(FVector::ZeroVector, 5); m_bUseRecursiveBezier = false; } // Called when the game starts or when spawned void AActor_BezierCuve::BeginPlay() { Super::BeginPlay(); m_points[0] = point0->GetComponentLocation(); m_points[1] = point1->GetComponentLocation(); m_points[2] = point2->GetComponentLocation(); m_points[3] = point3->GetComponentLocation(); m_points[4] = point4->GetComponentLocation(); if (!m_bUseRecursiveBezier) naive_bezier(); else bezier(); } // Called every frame void AActor_BezierCuve::Tick(float DeltaTime) { Super::Tick(DeltaTime); } // 多項式 void AActor_BezierCuve::naive_bezier() { FVector& p_0 = m_points[0]; FVector& p_1 = m_points[1]; FVector& p_2 = m_points[2]; FVector& p_3 = m_points[3]; FVector& p_4 = m_points[4]; for (double t = 0.0; t <= 1.0; t += 0.001) { auto point = std::pow(1 - t, 4) * p_0 + 4 * t * std::pow(1 - t, 3) * p_1 + 6 * std::pow(t, 2) * std::pow((1 - t), 2) * p_2 + 4 * std::pow(t, 3) * (1 - t) * p_3 + std::pow(t, 4) * p_4; DrawDebugPoint(GetWorld(), point, 2.0f, FColor::Green,true,5.0f); //UKismetSystemLibrary::PrintString(GetWorld(), point.ToString()); } } void AActor_BezierCuve::bezier() { for (double t = 0.0; t <= 1.0; t += 0.001) { FVector point = recursive_bezier(m_points, t); DrawDebugPoint(GetWorld(), point, 2.0f, FColor(10,214,255,255),true,5.0f); } } // De Casteljau 算法，遞歸 FVector AActor_BezierCuve::recursive_bezier(TArray<FVector>& points, float t) { if (points.Num() < 3) { return (1 - t) * points[0] + t * points[1]; } TArray<FVector> newPoint; for (int i = 0; i < points.Num() - 1; i++) { newPoint.Add((1 - t) * points[i] + t * points[i + 1]); } return recursive_bezier(newPoint, t); }