A Cardinal spline( sometimes called Canonical spline )consists of a sequence of curves. Consider a single Cardinal segment:<\/p>\n
\\[
\nF(u) = C_0 + uC_1 + u^2C_2 + u^3C_3
\n\\]<\/p>\n
Suppose it starts at \\(F(0) = P_i\\) and ends at \\(F(1)=P_{i+1}\\). We don’t need to provide the tangent for each point because it can be calculated from two ajacent control points. The constrant equations are very similar to the Hermite spline:<\/p>\n
\\begin{align}
\nP_i &= F(0)\u00a0 = C_0 \\\\
\nP_{i+1} &= F(1)\u00a0 =\u00a0 C_0 + C_1 + C_2 + C_3 \\\\
\nP’_i &= k(P_{i+1} – P_{i-1}) = F'(0) = C_1 \\\\
\nP’_{i+1} &= k(P_{i+2} – P_i) = F'(1) = C_1 + 2C_2 + 3C_3
\n\\end{align}<\/p>\n
The parameter k is a tension parameter that must be in the interval [0,1]. Now, \\(C\\) can be represented as:<\/p>\n
\\begin{align}
\nC_0 &= P_i \\\\
\nC_1 &= -kP_{i-1} + kP_{i+1} \\\\
\nC_2 &= 2kP_{i-1} + (k-3)P_i + (3-2k)P_{i+1} – kP_{i+2} \\\\
\nC_3 &= -kP_{i-1} + (2-k)P_i + (k-2)P_{i+1} + kP_{i+2}
\n\\end{align}<\/p>\n
Then we get the Cardinal matrix \\(A\\) and \\(C\\) can be represented as:<\/p>\n
\\begin{align}
\n\\begin{pmatrix} C_0 \\\\ C_1 \\\\ C_2 \\\\ C_3 \\end{pmatrix} =
\n\\begin{pmatrix}
\n0 & 1 & 0 & 0 \\\\
\n-k & 0 & k & 0 \\\\
\n2k & k-3 & 3-2k & -k \\\\
\n-k & 2-k & k-2 & k
\n\\end{pmatrix}
\n\\begin{pmatrix} P_{i-1} \\\\ P_i \\\\ P_{i+1} \\\\ P_{i+2} \\end{pmatrix}
\n\\end{align}<\/p>\n
Now we can derive the Cardinal basis:<\/p>\n
\\[
\n\\eta = \\left \\{ -ku + 2ku^2 – ku^3, ~~~\u00a0 1 + (k-3)u^2 + (2-k)u^3, ~~~\u00a0 ku + (3-2k)u^2 + (k-2)u^3,~~~\u00a0 -ku^2 + ku^3 \\right \\}
\n\\]<\/p>\n
Catmull-Rom spline is a special case of a Cardinal spline. Choosing tension parameter \\(k = \\frac 1 2\\) yields the basis:<\/p>\n
\\[
\n\\eta = \\left \\{ -\\frac 1 2 u + u^2 – \\frac 1 2 u^3, ~~~\u00a0 1 -\\frac 5 2 u^2 + \\frac 3 2 u^3, ~~~\u00a0 \\frac 1 2 u + 2u^2 -\\frac 3 2 u^3,~~~\u00a0 -\\frac 1 2 u^2 + \\frac 1 2 u^3 \\right \\}
\n\\]<\/p>\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
A Cardinal spline( sometimes called Canonical spline )consists of a sequence of curves. Consider a single Cardinal segment: \\[ F(u) = C_0 + uC_1 + u^2C_2 + u^3C_3 \\] Suppose it starts at \\(F(0) = P_i\\) and ends at \\(F(1)=P_{i+1}\\). We don’t need to provide the tangent for each point because it can be calculated […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-481","post","type-post","status-publish","format-standard","hentry","category-graphics"],"_links":{"self":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/481","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/comments?post=481"}],"version-history":[{"count":0,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/posts\/481\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/media?parent=481"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/categories?post=481"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.cgdev.net\/blog\/wp-json\/wp\/v2\/tags?post=481"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}