My REU project was about this topic.

We found some interesting things about when osculating cubics are unique (plot twist: not always), as well as a formula for the osculating conic of a smooth plane curve (at a non-flex point). Here is the formula (copied from a software package I wrote, sorry for the mess),

Given F(x,y):=(higher order terms) +a*x^4+b*x^3*y+c*x^2*y^2+d*x*y^3+e*y^4+f*x^3+g*x^2*y+h*x*y^2+i*y^3+j*x^2+k*x*y+l*y^2+m*x+n*y

 

The osculating conic of V(F) at (0,0) is given by

 

OscConic(a,b,c,d,e,f,g,h,i,j,k,l,m,n)=m*x+n*y+(j*x^2+k*x*y+l*y^2)+(-((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)^2)/(j*n^2-k*m*n+l*m^2)^3*(m*x+n*y)^2+(a*n^4-b*m*n^3+c*m^2*n^2+(-d)*m^3*n+e*m^4)/(j*n^2-k*m*n+l*m^2)^2*(m*x+n*y)^2+(((x*(k*m-2*j*n)+y*(2*l*m-k*n))*((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)-(x*(3*f*n^2-2*g*m*n+h*m^2)+y*(g*n^2-2*h*m*n+3*i*m^2))*(j*n^2-k*m*n+l*m^2))*(m*x+n*y))/(j*n^2-k*m*n+l*m^2)^2

The formula comes from the paper we wrote (look at Lemma 2.22 and 2.24) here

Points of Ninth Order on Cubic Curves

My REU project was about this topic.

We found some interesting things about when osculating cubics are unique (plot twist: not always), as well as a formula for the osculating conic of a smooth plane curve (at a non-flex point). Here is the formula (copied from a software package I wrote, sorry for the mess),

Given F(x,y):=(higher order terms) +a*x^4+b*x^3*y+c*x^2*y^2+d*x*y^3+e*y^4+f*x^3+g*x^2*y+h*x*y^2+i*y^3+j*x^2+k*x*y+l*y^2+m*x+n*y

The osculating conic of V(F) at (0,0) is given by

OscConic(a,b,c,d,e,f,g,h,i,j,k,l,m,n)=m*x+n*y+(j*x^2+k*x*y+l*y^2)+(-((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)^2)/(j*n^2-k*m*n+l*m^2)^3*(m*x+n*y)^2+(a*n^4-b*m*n^3+c*m^2*n^2+(-d)*m^3*n+e*m^4)/(j*n^2-k*m*n+l*m^2)^2*(m*x+n*y)^2+(((x*(k*m-2*j*n)+y*(2*l*m-k*n))*((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)-(x*(3*f*n^2-2*g*m*n+h*m^2)+y*(g*n^2-2*h*m*n+3*i*m^2))*(j*n^2-k*m*n+l*m^2))*(m*x+n*y))/(j*n^2-k*m*n+l*m^2)^2

The formula comes from the paper we wrote (look at Lemma 2.22 and 2.24) here

Points of Ninth Order on Cubic Curves

My REU project was about this topic.

We found some interesting things about when osculating cubics are unique (plot twist: not always), as well as a formula for the osculating conic of a smooth plane curve (at a non-flex point). Here is the formula (copied from a software package I wrote, sorry for the mess),

Given F(x,y):=(higher order terms) +ax^4+bx^3y+cx^2y^2+dxy^3+ey^4+fx^3+gx^2y+hxy^2+iy^3+jx^2+kxy+ly^2+mx+ny

The osculating conic of V(F) at (0,0) is given by

OscConic(a,b,c,d,e,f,g,h,i,j,k,l,m,n)=mx+ny+(jx^2+kxy+ly^2)+(-((-f)n^3+gmn^2+(-h)m^2n+im^3)^2)/(jn^2-kmn+lm^2)^3*(mx+ny)^2+(an^4-bmn^3+cm^2n^2+(-d)m^3n+em^4)/(jn^2-kmn+lm^2)^2*(mx+ny)^2+(((x*(km-2jn)+y(2lm-kn))((-f)n^3+gmn^2+(-h)m^2n+im^3)-(x*(3fn^2-2gmn+hm^2)+y*(gn^2-2hmn+3im^2))(jn^2-kmn+lm^2))(mx+ny))/(jn^2-kmn+lm^2)^2

The formula comes from the paper we wrote (look at Lemma 2.22 and 2.24) here

Points of Ninth Order on Cubic Curves

My REU project was about this topic.

We found some interesting things about when osculating cubics are unique (plot twist: not always), as well as a formula for the osculating conic of a smooth plane curve (at a non-flex point). Here is the formula (copied from a software package I wrote, sorry for the mess),

Given F(x,y):=(higher order terms) +a*x^4+b*x^3*y+c*x^2*y^2+d*x*y^3+e*y^4+f*x^3+g*x^2*y+h*x*y^2+i*y^3+j*x^2+k*x*y+l*y^2+m*x+n*y

The osculating conic of V(F) at (0,0) is given by

OscConic(a,b,c,d,e,f,g,h,i,j,k,l,m,n)=m*x+n*y+(j*x^2+k*x*y+l*y^2)+(-((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)^2)/(j*n^2-k*m*n+l*m^2)^3*(m*x+n*y)^2+(a*n^4-b*m*n^3+c*m^2*n^2+(-d)*m^3*n+e*m^4)/(j*n^2-k*m*n+l*m^2)^2*(m*x+n*y)^2+(((x*(k*m-2*j*n)+y*(2*l*m-k*n))*((-f)*n^3+g*m*n^2+(-h)*m^2*n+i*m^3)-(x*(3*f*n^2-2*g*m*n+h*m^2)+y*(g*n^2-2*h*m*n+3*i*m^2))*(j*n^2-k*m*n+l*m^2))*(m*x+n*y))/(j*n^2-k*m*n+l*m^2)^2

The formula comes from the paper we wrote (look at Lemma 2.22 and 2.24) here

Points of Ninth Order on Cubic Curves