【UE4】GAMES101 图形学作业4：贝塞尔曲线

总览

• Bézier 曲线是一种用于计算机图形学的参数曲线。

  

• 在本次作业中，你需要实现de Casteljau 算法来绘制由4 个控制点表示的Bézier 曲线(当你正确实现该算法时，你可以支持绘制由更多点来控制的Bézier 曲线)。

• 你需要修改的函数在提供的main.cpp 文件中。

  • bezier：该函数实现绘制Bézier 曲线的功能。
    
    它使用一个控制点序列和一个OpenCV::Mat 对象作为输入，没有返回值。它会使t 在0 到1 的范围内进行迭代，并在每次迭代中使t 增加一个微小值。对于每个需要计算的t，将调用另一个函数recursive_bezier，然后该函数将返回在Bézier 曲线上t处的点。最后，将返回的点绘制在OpenCV ：：Mat 对象上。
  • recursive_bezier：该函数使用一个控制点序列和一个浮点数t 作为输入，实现de Casteljau 算法来返回Bézier 曲线上对应点的坐标。

实现

• naive_bezier

  • 数学公式

    

  • 代码

    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 算法说明如下：

    1. 考虑一个p0, p1, ... pn 为控制点序列的Bézier 曲线。首先，将相邻的点连接起来以形成线段。
    2. 用t : (1 − t) 的比例细分每个线段，并找到该分割点。
    3. 得到的分割点作为新的控制点序列，新序列的长度会减少一。
    4. 如果序列只包含一个点，则返回该点并终止。否则，使用新的控制点序列并转到步骤1。使用[0,1] 中的多个不同的t 来执行上述算法，你就能得到相应的Bézier 曲线。

  • 代码

    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，递归
    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);
    }
    

• 最终效果
  
  

附录

所有代码

• Actor_BezierCuve.h

  点击查看代码

    ```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

  点击查看代码

  #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);
  
  }
  
  // 多项式
  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，递归
  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);
  }