3次スプライン曲線の作成方法 既知点

1 3 6 3 xn,yn .(xn,yn) (xn+1),(yn+1) 0 N N-1 6 0(x0,y0) (x1),(y1) 1(x1,y1) (x2),(y2) 2(x2,y2) (x3),(y3) 3(x3,y3) (x4),(y4) 4(x4,y4) (x5),(y5) 3 ( (x0,y0) (x1,y1) ) 3 an , bn , cn , dn . (x1,y1)(x2,y2)(x4,y4)(x5,y5)(x3,y3) nnnnnnnndxxcxxbxxaS 23 0002003000dxxcxxbxxaS 1112113111dxxcxxbxxaS 2222223222dxxcxxbxxaS 4442443444dxxcxxbxxaS 3332333333dxxcxxbxxaS 1 dn x=xn (x-xn)=0 -1bn Sn(x) x=xn (x-xn)

2 =0 -2an -3cn -1,2,3 cn x y nnnnnnnnndxxcxxbxxaSy 23nnyd nnnnnnnndxxcxxbxxaS 23nxxf )()1()(' nxnxfnnnnnnncxxbxxaxxS )(2)(3)('2nnnnnbxxaxxS2)(6)(" nnnnnbxxaxxS2)(6)(" )(6)2)("(11nnnnnxxbxSa )(6))(")("(11nnnnnxxxSxSa 123 nnnnnnnnydxxcxxbxxa112131111)()(2)("))(( )(62)(" nnnnnnnnnnnnnnbnyyxxcxxxSxxxxxxbxSnnnnnb xxaxxS2)(6)(" 2)("nnnxxSb 2213111111)(2)("))(()(6)(")(")(nnnnnnnnn nnnnnnnxxxSxxxxxxxSxSyyxxc 3 xn+1 213111111)(2)("))(()(6)(")(")(nnnnnnnnnn nnnnnnxxxSxxxxxxxSxSyyxxc nnnnnnnnnnnnnnnnxxxxxSxxxxxxxSxSyyc 121311111)(2)("))(()(6)(")("6)(6))((111x xSxxSS 363611111xSxSxSxS 6221111xSxSxSxS acbcadbddcba ))(()2)((6111 SSxx ))(")("2)((61)()(1111xSxSxxxxyycnnnnnnnn n )(")('111 nnnnxSxSnnnnnnncxxbxxaxxS )(2)(3)('2nnnnnnnnnnnnnnnnxxxxxSxxxxxxxS xSyyc 1213111111)(2)("))(()(6)(")(" 2))((")(6)()(")

3 ("1121111nnnnnnnnnnnnnnxxxSxxxxxSxSxxyyc 4xn+1 x-xn 0 an,bn,cn Sn" Sn+1" Sn+2" 1121)(2)(3 nnnnnnnnccxxbxxa)(6)(")("11nnnnnnnxxxSxS a 2)("nnnxSb ))(")("2)((61)()(1111nnnnnnnnnnnxSxSxxxx yyc ))(")("2)((61)1()(211122121nnnnnnnnnnnxS xSxxxxyyc ))(2)("(2))()(6)(")("(312111nnnnnnnnnnnn xxxSxxxxxSxS ))(")("2)((61)1()(21112212nnnnnnnnnnxSxS xxxxyy ))(")("2)((61))(("))(2)(")("(11111nnnnnn nnnnnnnnnnxSxSxxxxxSxxxSxS )()()1()(11212nnnnnnnnxxyyxxyy )(")(31))((")(2)(")(2)("11111nnnnnnnnnnn nnnnnxSxxxxxSxxxSxxxS )()()1()(11212nnnnnnnnxxyyxxyy )))(")("2)((61)()((1111nnnnnnnnnnxSxSxxx xyy ))(")("2)((6121112nnnnnnxSxSxx)(")(31))( ")(61111211 nnnnnnnnxSxxxSxx )(")(61212nnnnxSxx5 6 6 0(x0,y0) (x1),(y1) n=01(x1,y1) (x2),(y2) n=12(x2,y2) (x3),(y3) n=2 )x(x31)x(x2)x(x)(x"Sn1nn1nn1nnn)()()1()( 11212nnnnnnnnxxyyxxyy )(31)("6)()("1211nnnnnnnnxxxSxxxS)()()1( )(11212nnnnnnnnxxyyxxyy )()()1()(6)(")()(")(2)(")(112122121121nn nnnnnnnnnnnnnnnnnnxxyyxxyyxSxxxSxxxSxx )()()1()(6)(")()(")(2)(")(112122121121nn nnnnnnnnnnnnnnnnnnxxyyxxyyxSxxxSxxxSxx )()()1()(6)(")()(")(2)(")

4 (112122121121nnnnnnnnnnnnnnnnnnnnxxyyxxy yxSxxxSxxxSxx )()()1()(6)(")()(")(2)(")(112122121121nn nnnnnnnnnnnnnnnnnnxxyyxxyyxSxxxSxxxSxx )(")(61)(31)(61)(21)("21212111nnnnnnnnnn nnxSxxxxxxxxxS )(")(61212nnnnxSxx63(x3,y3) (x4),(y4) n=34(x4,y4) (x5),(y5) n=4 0 N 0 N-2 S0"(x)=SN-1"(x)=0 0 N-2 n=0 n=1 N-4 )()()1()(6)(")()(")(2)(")(112122121121nn nnnnnnnnnnnnnnnnnnxxyyxxyyxSxxxSxxxSxx )()()1()(6)(")()(")(2)(")(112122121121nn nnnnnnnnnnnnnnnnnnxxyyxxyyxSxxxSxxxSxx )()()1()(6)(")()(")(2)(")(112122121121nn nnnnnnnnnnnnnnnnnnxxyyxxyyxSxxxSxxxSxxnn nxxh 1nnnnnnnnxxxxxxhh 21121 1116nnnnnnnhyyhyyk121101"")(2kShShh 23221211"")(2"kShShhSh 232322")(2"kShhSh 121101"")(2kShShh 12111"")(2" nnnnnnnnkShShhSh323233")(2" NNNNNNkShhSh7n=N-3 2 x,y | | 2 3 4 Excel maximax Excel MDETERM 3213212322121101""")

5 (20)(20)(2kkkSSShhhhhhhhhh 323233")(2" NNNNNNkShhSh 45124342543yxyxyx 2122211211bbyxaaaa 22211211222121aaaaababx 22221212221111aaaababay 2112221122211211aaaaaaaa 89 hn kn " )(20)(20)(2)(2)(20"232212110123232122111 hhhhhhhhhhhhhkhhhkhkS )(20)(20)(2)(200)(2"2322121101233221101h hhhhhhhhhhhkhkhkhhS )(20)(20)(20)(2)(2"232212110132212111013 hhhhhhhhhhkhkhhhkhhhS nnnxxh 1 " = " hn,kn + = + = + = a11a12a13a21a22a23a31a32a33 S2"=S3"= = =S1"= =11ui1=ai1/a11aji = aji - ui1 * a1jbi=bi-ui1*b1a11a12a13a21-u11*a11a22-u 11*a12a23-u11*a13a31-u31*a11a32-u31*a12a 33-u31*a13 ai ' a11'a12'a13'0a22'a23'0a32'a33' 3 2 ui2'=ai2'/a22'aij'=aji'-ui2*a2j'bi=bi-ui 1*bia11'a12'a13'0a22'a23'0a32'=a32'-ui2' *a22'a33'=a33'-ui2'*a23' a11'a12'a13'0a22'a23'00a33'=a33'-ui2'*a2 3'.)

6 A11'*a22'*a33'=a33'-ui2'*a23' .12ui1=ai1/a11aji = aji - ui1 * a1jbi=bi-ui1* '=aji'-ui2*a2j'bi=bi-ui1*bi123knui2'=ai2 '/a22' S3"= "= ( + * ) = "= ( +0* + * ) = " S3" an,bn,cn,dn )(6))(")("(11nnnnnxxxSxSa nnyd 2)("nnnxSb ))(")("2)((61)()(1111nnnnnnnnnnnxSxSxxxx yyc 13xySn" an,bn,cn,dn x 4 xySn" nnnnnnnnndxxcxxbxxaSy 2315 1000120014001600180020002200240026002800 3000200025003000350040004500500055006000 16