平滑曲线生成：贝塞尔曲线拟合

//count为插入点数, outPoints为输出点集合，长度为count + 2(含首尾)
void BezierHelper::parseBezier(const Ogre::Vector2& start, const Ogre::Vector2& end,    
    const Ogre::Vector2& control1, const Ogre::Vector2& control2, int count, std::vector<Ogre::Vector2>& outPoints)
{
    if(count < 0) count = Bezier3DefaultPointNum;

    outPoints.push_back(start);
    for(int i = 1; i<=count; i++)
    {
        double st = (double) i/(count+1);
        double dt = (1-st);

        //二次项
        double st2 = st * st; 
        double dt2 = dt * dt;

        double t0 = dt * dt2;
        double t1 = dt2 * st * 3; 
        double t2 = dt * st2 * 3;
        double t3 = st * st2; 

        outPoints.push_back(start * t0 + control1 * t1 + control2 * t2 + end * t3);
    }

    outPoints.push_back(end);
}

void BezierHelper::_parseBezier(const Ogre::Vector2& start, const Ogre::Vector2& end,    
    const Ogre::Vector2& control1, const Ogre::Vector2& control2, int count, std::vector<Ogre::Vector2>& outPoints)
{
    if(count < 0) count = Bezier3DefaultPointNum;

    outPoints.push_back(start);
    for(int i = 1; i<=count; i++)
    {
        double st = (double) i/(count+1);
        double dt = (1-st);

        //二次项
        double st2 = st * st; 
        double dt2 = dt * dt;

        double t0 = dt * dt2;
        double t1 = dt2 * st * 3; 
        double t2 = dt * st2 * 3;
        double t3 = st * st2; 

        outPoints.push_back(start * t0 + control1 * t1 + control2 * t2 + end * t3);
    }
}

void BezierHelper::parsePolygon(const std::vector<Ogre::Vector2>& points, int count, std::vector<Ogre::Vector2>& outPoints, double ratio)
{
    std::vector<Ogre::Vector2>::size_type pointsSize = points.size();
    if(pointsSize < 3 )//插值至少需要三点
    {
        for(int i = 0; i<pointsSize; i++) 
        {
            outPoints.push_back(points[i]);
        }
    }
    else
    {
        Ogre::Vector2  control1, control2;    //顶点对应的前后控制点
        Ogre::Vector2 firstControl;
        Ogre::Vector2 lastControl;

        //首尾
        getControlPoint(points[pointsSize-1], points[pointsSize-2], points[0], firstControl, lastControl, ratio); 
        getControlPoint(points[0], points[pointsSize-1], points[1], control1, control2, ratio); 
        _parseBezier(points[pointsSize-1], points[0], lastControl, control1, count, outPoints);

        for(int i = 1; i<pointsSize-1; i++) //根据中间各点，生成与前一点连线
        {
            lastControl = control2;
            getControlPoint(points[i], points[i-1], points[i+1], control1, control2, ratio);
            _parseBezier(points[i-1], points[i], lastControl, control1, count, outPoints);

        }

        parseBezier(points[pointsSize-2], points[pointsSize-1], control2, firstControl, count, outPoints); //末端
        outPoints.pop_back();
    }
}

#ifndef __Vector2_H__
#define __Vector2_H__
#include <assert.h>

namespace Ogre
{

    class Vector2
    {
    public:
        double x, y;

    public:
        inline Vector2()
        {
        }

        inline Vector2(const double fX, const double fY )
            : x( fX ), y( fY )
        {
        }

        inline Vector2& operator = ( const Vector2& rkVector )
        {
            x = rkVector.x;
            y = rkVector.y;

            return *this;
        }

        inline Vector2& operator = ( const double fScalar)
        {
            x = fScalar;
            y = fScalar;

            return *this;
        }

        inline bool operator == ( const Vector2& rkVector ) const
        {
            return ( x == rkVector.x && y == rkVector.y );
        }

        inline bool operator != ( const Vector2& rkVector ) const
        {
            return ( x != rkVector.x || y != rkVector.y  );
        }

        // arithmetic operations
        inline Vector2 operator + ( const Vector2& rkVector ) const
        {
            return Vector2(
                x + rkVector.x,
                y + rkVector.y);
        }

        inline Vector2 operator - ( const Vector2& rkVector ) const
        {
            return Vector2(
                x - rkVector.x,
                y - rkVector.y);
        }

        inline Vector2 operator * ( const double fScalar ) const
        {
            return Vector2(
                x * fScalar,
                y * fScalar);
        }

        inline Vector2 operator * ( const Vector2& rhs) const
        {
            return Vector2(
                x * rhs.x,
                y * rhs.y);
        }

        inline Vector2 operator / ( const double fScalar ) const
        {
            assert( fScalar != 0.0 );

            double fInv = 1.0f / fScalar;

            return Vector2(
                x * fInv,
                y * fInv);
        }

        inline Vector2 operator / ( const Vector2& rhs) const
        {
            return Vector2(
                x / rhs.x,
                y / rhs.y);
        }

        inline const Vector2& operator + () const
        {
            return *this;
        }

        inline Vector2 operator - () const
        {
            return Vector2(-x, -y);
        }

        // overloaded operators to help Vector2
        inline friend Vector2 operator * ( const double fScalar, const Vector2& rkVector )
        {
            return Vector2(
                fScalar * rkVector.x,
                fScalar * rkVector.y);
        }

        inline friend Vector2 operator / ( const double fScalar, const Vector2& rkVector )
        {
            return Vector2(
                fScalar / rkVector.x,
                fScalar / rkVector.y);
        }

        inline friend Vector2 operator + (const Vector2& lhs, const double rhs)
        {
            return Vector2(
                lhs.x + rhs,
                lhs.y + rhs);
        }

        inline friend Vector2 operator + (const double lhs, const Vector2& rhs)
        {
            return Vector2(
                lhs + rhs.x,
                lhs + rhs.y);
        }

        inline friend Vector2 operator - (const Vector2& lhs, const double rhs)
        {
            return Vector2(
                lhs.x - rhs,
                lhs.y - rhs);
        }

        inline friend Vector2 operator - (const double lhs, const Vector2& rhs)
        {
            return Vector2(
                lhs - rhs.x,
                lhs - rhs.y);
        }

        // arithmetic updates
        inline Vector2& operator += ( const Vector2& rkVector )
        {
            x += rkVector.x;
            y += rkVector.y;

            return *this;
        }

        inline Vector2& operator += ( const double fScaler )
        {
            x += fScaler;
            y += fScaler;

            return *this;
        }

        inline Vector2& operator -= ( const Vector2& rkVector )
        {
            x -= rkVector.x;
            y -= rkVector.y;

            return *this;
        }

        inline Vector2& operator -= ( const double fScaler )
        {
            x -= fScaler;
            y -= fScaler;

            return *this;
        }

        inline Vector2& operator *= ( const double fScalar )
        {
            x *= fScalar;
            y *= fScalar;

            return *this;
        }

        inline Vector2& operator *= ( const Vector2& rkVector )
        {
            x *= rkVector.x;
            y *= rkVector.y;

            return *this;
        }

        inline Vector2& operator /= ( const double fScalar )
        {
            assert( fScalar != 0.0 );

            double fInv = 1.0f / fScalar;

            x *= fInv;
            y *= fInv;

            return *this;
        }

        inline Vector2& operator /= ( const Vector2& rkVector )
        {
            x /= rkVector.x;
            y /= rkVector.y;

            return *this;
        }

        /** Returns the length (magnitude) of the vector.
            @warning
                This operation requires a square root and is expensive in
                terms of CPU operations. If you don‘t need to know the exact
                length (e.g. for just comparing lengths) use squaredLength()
                instead.
        */
        inline double length () const
        {
            return sqrt( x * x + y * y );
        }

        /** Returns the square of the length(magnitude) of the vector.
            @remarks
                This  method is for efficiency - calculating the actual
                length of a vector requires a square root, which is expensive
                in terms of the operations required. This method returns the
                square of the length of the vector, i.e. the same as the
                length but before the square root is taken. Use this if you
                want to find the longest / shortest vector without incurring
                the square root.
        */
        inline double squaredLength () const
        {
            return x * x + y * y;
        }

        /** Returns the distance to another vector.
            @warning
                This operation requires a square root and is expensive in
                terms of CPU operations. If you don‘t need to know the exact
                distance (e.g. for just comparing distances) use squaredDistance()
                instead.
        */
        inline double distance(const Vector2& rhs) const
        {
            return (*this - rhs).length();
        }

        /** Returns the square of the distance to another vector.
            @remarks
                This method is for efficiency - calculating the actual
                distance to another vector requires a square root, which is
                expensive in terms of the operations required. This method
                returns the square of the distance to another vector, i.e.
                the same as the distance but before the square root is taken.
                Use this if you want to find the longest / shortest distance
                without incurring the square root.
        */
        inline double squaredDistance(const Vector2& rhs) const
        {
            return (*this - rhs).squaredLength();
        }

        /** Calculates the dot (scalar) product of this vector with another.
            @remarks
                The dot product can be used to calculate the angle between 2
                vectors. If both are unit vectors, the dot product is the
                cosine of the angle; otherwise the dot product must be
                divided by the product of the lengths of both vectors to get
                the cosine of the angle. This result can further be used to
                calculate the distance of a point from a plane.
            @param
                vec Vector with which to calculate the dot product (together
                with this one).
            @return
                A float representing the dot product value.
        */
        inline double dotProduct(const Vector2& vec) const
        {
            return x * vec.x + y * vec.y;
        }

        /** Normalises the vector.
            @remarks
                This method normalises the vector such that it‘s
                length / magnitude is 1. The result is called a unit vector.
            @note
                This function will not crash for zero-sized vectors, but there
                will be no changes made to their components.
            @return The previous length of the vector.
        */

        inline double normalise()
        {
            double fLength = sqrt( x * x + y * y);

            // Will also work for zero-sized vectors, but will change nothing
            // We‘re not using epsilons because we don‘t need to.
            // Read http://www.ogre3d.org/forums/viewtopic.php?f=4&t=61259
            if ( fLength > double(0.0f) )
            {
                double fInvLength = 1.0f / fLength;
                x *= fInvLength;
                y *= fInvLength;
            }

            return fLength;
        }

        /** Returns a vector at a point half way between this and the passed
            in vector.
        */
        inline Vector2 midPoint( const Vector2& vec ) const
        {
            return Vector2(
                ( x + vec.x ) * 0.5f,
                ( y + vec.y ) * 0.5f );
        }

        /** Returns true if the vector‘s scalar components are all greater
            that the ones of the vector it is compared against.
        */
        inline bool operator < ( const Vector2& rhs ) const
        {
            if( x < rhs.x && y < rhs.y )
                return true;
            return false;
        }

        /** Returns true if the vector‘s scalar components are all smaller
            that the ones of the vector it is compared against.
        */
        inline bool operator > ( const Vector2& rhs ) const
        {
            if( x > rhs.x && y > rhs.y )
                return true;
            return false;
        }

        /** Sets this vector‘s components to the minimum of its own and the
            ones of the passed in vector.
            @remarks
                ‘Minimum‘ in this case means the combination of the lowest
                value of x, y and z from both vectors. Lowest is taken just
                numerically, not magnitude, so -1 < 0.
        */
        inline void makeFloor( const Vector2& cmp )
        {
            if( cmp.x < x ) x = cmp.x;
            if( cmp.y < y ) y = cmp.y;
        }

        /** Sets this vector‘s components to the maximum of its own and the
            ones of the passed in vector.
            @remarks
                ‘Maximum‘ in this case means the combination of the highest
                value of x, y and z from both vectors. Highest is taken just
                numerically, not magnitude, so 1 > -3.
        */
        inline void makeCeil( const Vector2& cmp )
        {
            if( cmp.x > x ) x = cmp.x;
            if( cmp.y > y ) y = cmp.y;
        }

        /** Generates a vector perpendicular to this vector (eg an ‘up‘ vector).
            @remarks
                This method will return a vector which is perpendicular to this
                vector. There are an infinite number of possibilities but this
                method will guarantee to generate one of them. If you need more
                control you should use the Quaternion class.
        */
        inline Vector2 perpendicular(void) const
        {
            return Vector2 (-y, x);
        }

        /** Calculates the 2 dimensional cross-product of 2 vectors, which results
            in a single floating point value which is 2 times the area of the triangle.
        */
        inline double crossProduct( const Vector2& rkVector ) const
        {
            return x * rkVector.y - y * rkVector.x;
        }

        /** Returns true if this vector is zero length. */
        inline bool isZeroLength(void) const
        {
            double sqlen = (x * x) + (y * y);
            return (sqlen < (1e-06 * 1e-06));

        }

        /** As normalise, except that this vector is unaffected and the
            normalised vector is returned as a copy. */
        inline Vector2 normalisedCopy(void) const
        {
            Vector2 ret = *this;
            ret.normalise();
            return ret;
        }

        /** Calculates a reflection vector to the plane with the given normal .
        @remarks NB assumes ‘this‘ is pointing AWAY FROM the plane, invert if it is not.
        */
        inline Vector2 reflect(const Vector2& normal) const
        {
            return Vector2( *this - ( 2 * this->dotProduct(normal) * normal ) );
        }

 
    };

}
#endif