Hermite Spline implementation from the scratch.

1. Mobile Game Programming:
Hermite Spline
jintaeks@dongseo.ac.kr
Division of Digital Contents, DongSeo University.
October 2016

2. Type of functions
Explicit function
y=
2
𝑅2 − 𝑥2 or y=-
2
𝑅2 − 𝑥2
Implicit function
x2+y2=R2
Parametric representation
x=Rcosθ, y=Rsinθ (0≤ θ < 2π)
2

3. Order of continuity
 The various order of parametric continuity can be described
as follows:[5]
 C−1: curves include discontinuities
 C0: curves are joined
 C1: first derivatives are continuous
 C2: first and second derivatives are continuous
 Cn: first through nth derivatives are continuous
3

4. Spline
 In mathematics, a spline is a
numeric function that is piecewise-defined
by polynomial functions.
– A flexible device which can be bent to the
desired shape is known as a flat spline.
 In interpolation problems, spline
interpolation is often preferred
to polynomial interpolation because it
yields similar results to interpolating with
higher degree polynomials while avoiding
instability.
 The most commonly used splines
are cubic splines.
4

5. Ex) Quadratic spline
5

6. Ex) Cubic spline
6

7. Definition of spline
 A spline is a piecewise-polynomial real function
S:[a,b]  R
 on an interval [a,b] composed of k subintervals [ti-1,ti] with
a=t0<t1< … <tk-1<tk=b.
 The restriction of S to an interval i is a polynomial
Pi:[ti-1,ti]R,
 so that
S(t)=P1(t), t0<= t <= t1,
S(t)=P2(t), t1<= t <= t2,
…
S(t)=Pk(t), tk-1<= t <= tk,
 The highest order of the polynomials Pi(t) is said to be
the order of the spline S.
7

8. Cubic spline
 In numerical analysis, a cubic spline is a spline where each
piece is a third-degree polynomial.
 A cubic Hermite spline Specified by its values and
first derivatives at the end points of the
corresponding domain interval.
8

9. Unit interval (0,1)
 On the interval (0,1), given a starting point p0 at t = 0 and an
ending point p1 at t = 1 with starting tangent m0 at t = 0 and
ending tangent m1 at t = 1, the polynomial can be defined by
 where t ∈ [0,1].
9

10.  a point Q(t) on a curve at parameter t can be defined a cubic
equations for each value:
Q(t)=
𝑥(𝑡)
𝑦(𝑡)
=
𝑎 𝑥 𝑡3 + 𝑏𝑥𝑡2 + 𝑐𝑥𝑡 + 𝑑𝑦
𝑎 𝑦 𝑡3 + 𝑏𝑦𝑡2 + 𝑐𝑦𝑡 + 𝑑𝑦
 We can define coefficient matrix C and parameter matrix T,
then Q(t) can be defined like this:
T=
𝑡3
𝑡2
𝑡
1
C=
𝑎 𝑥 𝑏 𝑥 𝑐 𝑥 𝑑 𝑥
𝑎 𝑦 𝑏 𝑦 𝑐 𝑦 𝑑 𝑦
Q(t)=
𝑥(𝑡)
𝑦(𝑡)
=𝐶‧𝑇
10

11.  We will decompose C with 2×4 Geometry Matrix G and 4×4
Basis Matrix M.
C=G‧M
Q(t)=
𝑥(𝑡)
𝑦(𝑡)
=𝑪‧𝑇 = 𝐺‧M‧T
=
𝑔1𝑥 𝑔2𝑥
𝑔3𝑥 𝑔4𝑥
𝑔1𝑦 𝑔2𝑦
𝑔3𝑦 𝑔4𝑦
𝑚11 𝑚12 𝑚13 𝑚14
𝑚21 𝑚22 𝑚23 𝑚24
𝑚31 𝑚32 𝑚33 𝑚34
𝑚41 𝑚42 𝑚43 𝑚44
𝑡3
𝑡2
𝑡
1
11

12. Geometry matrix G
 has an information about control data.
 in case of Hermite spline, it has two points and tangent
vectors for each point.
– P1, P2, R1, R2
Base matrix M
 a coefficient matrix for the blending functions.
 blending function means M‧T.
 the summation of all blending functions is 1.
– it means that spline curve calculates the weighted average for control
points.
12

13. Hermite Spline
 Specified by its values and first derivatives at the end points
of the corresponding domain interval
Geometry matrix G
G=
𝑃1𝑥 𝑃4𝑥 𝑅1𝑥 𝑅4𝑥
𝑃1𝑦 𝑃4𝑦 𝑅1𝑦 𝑅4𝑦
13

14. Find Base matrix M
 Four conditions to find Base matrix M.
 Begin point is (P1x, P1y), and End point is (P4x, P4y).
Q(0)=
𝑃1𝑥
𝑃1𝑦
Q(1)=
𝑃4𝑥
𝑃4𝑦
 Derivative at Begin point is (R1x, R1y), and derivative at End
point is (R4x, R4y).
Q'(0)=
𝑅1𝑥
𝑅1𝑦
Q'(1)=
𝑅4𝑥
𝑅4𝑦
14

15. for two end points
 Q(0)=
𝑃1𝑥
𝑃1𝑦
= 𝐺‧M‧T
=
𝑃1𝑥 𝑃4𝑥 𝑅1𝑥 𝑅4𝑥
𝑃1𝑦 𝑃4𝑦 𝑅1𝑦 𝑅4𝑦
𝑚11 𝑚12 𝑚13 𝑚14
𝑚21 𝑚22 𝑚23 𝑚24
𝑚31 𝑚32 𝑚33 𝑚34
𝑚41 𝑚42 𝑚43 𝑚44
𝑡3
𝑡2
𝑡
1
=𝐺‧M
0
0
0
1
 Q(1)=
𝑃4𝑥
𝑃4𝑦
= 𝐺‧M
1
1
1
1
15

16. for two tangent vector
 Q'(0)=
𝑅1𝑥
𝑅1𝑦
= 𝐺‧M‧T′
=
𝑃1𝑥 𝑃4𝑥 𝑅1𝑥 𝑅4𝑥
𝑃1𝑦 𝑃4𝑦 𝑅1𝑦 𝑅4𝑦
𝑚11 𝑚12 𝑚13 𝑚14
𝑚21 𝑚22 𝑚23 𝑚24
𝑚31 𝑚32 𝑚33 𝑚34
𝑚41 𝑚42 𝑚43 𝑚44
3𝑡2
2𝑡
1
0
=𝐺‧M
0
0
1
0
 Q'(1)=
𝑅4𝑥
𝑅4𝑦
= 𝐺‧M
3
2
1
0
16

17. Base matrix M

𝑃1𝑥 𝑃4𝑥 𝑅1𝑥 𝑅4𝑥
𝑃1𝑦 𝑃4𝑦 𝑅1𝑦 𝑅4𝑦
= 𝐺 = 𝐺‧𝑀
0 1 0 3
0 1 0 2
0 1 1 1
1 1 0 0
= 𝐺‧𝑀‧𝑀-1
0 1 0 3
0 1 0 2
0 1 1 1
1 1 0 0
=M-1
 M=
2 −3 0 1
−2 3 0 0
1 −2 1 0
1 −1 0 0
17

18. Blending function
 Q(t)=
𝑥(𝑡)
𝑦(𝑡)
= 𝐺‧M‧T = 𝐺‧𝐵
=(2t3−3t2+1)
𝑃1𝑥
𝑃1𝑦
+(-2t3+3t2)
𝑃4𝑥
𝑃4𝑦
+(t3-2t2+1)
𝑅1𝑥
𝑅1𝑦
+(t3-t2)
𝑅4𝑥
𝑅4𝑦
.
18
2 −3 0 1
−2 3 0 0
1 −2 1 0
1 −1 0 0
𝑡3
𝑡2
𝑡1
0

19. Meaning of tangent vector R
19
P1 P4
R4
R1

20. Implement Hermite Spline
class KHermiteCurve
{
public:
float m_a;
float m_b;
float m_c;
float m_d;
20

21. class KHermiteCurve
{
public:
/// constructor.
/// @param p1: begin point
/// @param p2: end point
/// @param v1: tangent vector at begin point
/// @param v2: tangent vector at end point
KHermiteCurve(float p1, float p2, float v1, float v2)
{
m_d = p1;
m_c = v1;
m_b = -3.0f*p1 + 3.0f*p2 - 2.0f*v1 - v2;
m_a = 2.0f*p1 - 2.0f*p2 + v1 + v2;
}//KHermiteCurve()
21

22. /// set coefficient of Hermite curve
inline void Construct(float p1, float p2, float v1, float v2)
{
m_d = p1;
m_c = v1;
m_b = -3.0f*p1 + 3.0f*p2 - 2.0f*v1 - v2;
m_a = 2.0f*p1 - 2.0f*p2 + v1 + v2;
}//Construct()
/// calculate first derivative on parameter u.
/// can be used to get tangent vector at u.
/// derivative of equation t^3 + t^2 + t + 1
inline float CalculateDxDu(float u) const
{
return 3.0f*m_a*u*u + 2.0f*m_b*u + m_c;
}//CalculateDxDu()
22

23. /// get value at parameter u.
inline float CalculateX(float u) const
{
float uu, uuu;
uu = u * u;
uuu = uu * u;
return m_a*uuu + m_b*uu + m_c*u + m_d;
}//CalculateX()
23

24. KHermiteSpline2
class KHermiteSpline2
{
private:
KHermiteCurve m_aHermiteCurve[2];
public:
/// constructor.
KHermiteSpline2(){}
KHermiteSpline2( const KVector& p0, const KVector& p1, const KVector& dp0, const
KVector& dp1 )
{
m_aHermiteCurve[ 0 ].Construct( p0.m_x, p1.m_x, dp0.m_x, dp1.m_x );
m_aHermiteCurve[ 1 ].Construct( p0.m_y, p1.m_y, dp0.m_y, dp1.m_y );
}//KHermiteSpline2()
24

25. inline void Construct( const KVector& p0, const KVector& p1, const KVector& dp0,
const KVector& dp1 )
{
m_aHermiteCurve[ 0 ].Construct( p0.m_x, p1.m_x, dp0.m_x, dp1.m_x );
m_aHermiteCurve[ 1 ].Construct( p0.m_y, p1.m_y, dp0.m_y, dp1.m_y );
}//Construct()
25

26. inline const KVector GetPosition( float u_ ) const
{
return KVector( m_aHermiteCurve[ 0 ].CalculateX( u_ ), m_aHermiteCurve[ 1
].CalculateX( u_ ), 0.f );
}//GetPosition()
inline const KVector GetTangent( float u_ ) const
{
return KVector( m_aHermiteCurve[ 0 ].CalculateDxDu( u_ ), m_aHermiteCurve[ 1
].CalculateDxDu( u_ ), 0.f );
}//GetTangent()
26

27. Draw a tangent line at u
const float u = 0.5f;
KVector position = spline.GetPosition(u);
KVector tangent = spline.GetTangent(u);
DrawLine(hdc, position, position + tangent * 100.f);
27

28. Combine Splines
28

29. Line in 3D space
 Given two 3D points P1 and P2, we can define the line that
passes through these points parametrically as
P(t ) = (1− t )P1 + tP2
 A ray is a line having a single endpoint S and extending to
infinity in a given direction V. Rays are typically expressed by
the parametric equation
P(t ) = S + tV
 Note that this equation is equivalent to previous equation if
we let S = P1 and V = P2 − P1.
29

30. Distance between a point and a line
30

31. Spline path tracing demo
31