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This pencil is usually made from non-toxic opaque wax (such as paraffin, beeswax, ceresin, carnauba or spermaceti wax) [1] that is similar to a crayon but stronger. [citation needed] Marks made by grease pencils are resistant to moisture and can usually be removed by rubbing the marked surface with a paper towel.
A crunode at the origin of the curve defined by (+) =. In mathematics, a crunode [1] (archaic; from Latin crux "cross" + node [2]) or node of an algebraic curve is a type of singular point at which the curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection.
The characteristic linear system of a family of curves on an algebraic surface Y for a curve C in the family is a linear system formed by the curves in the family that are infinitely near C. [ 4 ] In modern terms, it is a subsystem of the linear system associated to the normal bundle to C ↪ Y {\displaystyle C\hookrightarrow Y} .
A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending points along integral curves of the vector field. The local flow of a vector field is used to define the Lie derivative of tensor fields along the vector field.
Every elliptic curve is birationally equivalent to a curve of the Hesse pencil; this is the Hessian form of an elliptic curve. However, the parameters ( λ , μ {\displaystyle \lambda ,\mu } ) of the Hessian form may belong to an extension field of the field of definition of the original curve.
1. This is the depth of a node plus 1, although some [12] define it instead to be synonym of depth. A node's level in a rooted tree is the number of nodes in the path from the root to the node. For instance, the root has level 1 and any one of its adjacent nodes has level 2. 2. A set of all node having the same level or depth. [12] line
Using a black Royal Sovereign Chinagraph pencil I am exploring methods of blending, spreading, removing and fixing the black marks on paper, for artwork sketching and drawing. The black marks can be removed easily by washing in cold soapy water. That is not what would be expected from a wax, but is typical of an oil or grease I believe.
The central curve, the cardioid, has a cusp at the origin. The right curve has a crunode at the origin and the curve crosses itself to form a loop. If b 0 and b 1 are both 0 in the above expansion, but at least one of c 0, c 1, c 2 is not 0 then the origin is called a double point of the curve.