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Scalars, vectors and fields
Published in Alan Jeffrey, Mathematics, 2004
Now the equation of a plane is n.r=p, where r = xi+yj+zk is a general point on the plane, n is the unit normal to the plane, and p is its perpendicular distance from the origin. The point r0 = i+2j+k is a point on the plane so that n.r = n.r0( =p), which after substituting for n, r and r0 becomes (14i+13j+6k√401)·(xi+yj+zk)=(14i+23j+6k√401)·(i+2j+k), showing that the required equation is 14x+13y+6z=46.
The definition of time and different time systems
Published in Lucien Wald, Fundamentals of Solar Radiation, 2021
In geometry, the perpendicular to a surface is also called normal to the surface. The angle of incidence θ is equal to 0 when the incident rays are perpendicular to the plane. This is called normal incidence. Conversely, θ is equal to π/2 (90°) when the incident rays are parallel to the plane. This is called grazing incidence.
Limit Loads of Structures
Published in Lingyi Lu, Junbo Jia, Zhuo Tang, Structural Mechanics, 2022
Lingyi Lu, Junbo Jia, Zhuo Tang
In mathematics, a normal vector of a curve at a point is defined as a unit vector pointing towards the center of curvature. When the local region of a plastic hinge is enlarged, then it is differentiable. The normal vector at a plastic hinge is called a normal vector of the plastic hinge.
Stabilising visualisation by reducing camera movements in virtual colonoscopy methods
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2021
Mostafa Mohamed, Asem Ali, Salwa Elshazly, Aly Farag
We estimate the original colon centerline using the method presented in Hassouna and Farag (2008). To enforce all the points of the centerline to be collinear or coplanar, we segment the set of points of the centerline into subsets. Each subset contains consecutive points, which can be fitted into a plane or a line with an arbitrary maximum error. Define the ordered set of the centerline points as where is the number of centerline points. A point is a 3D point in the form . A plane is generally described by a normal vector and a distance so that for a point on the plane satisfies . We can write this as . Since the eigenvectors of the covariance matrix
Reconstruction of a CAD model from the deformed mesh using B-spline surfaces
Published in International Journal of Computer Integrated Manufacturing, 2019
Aicha Ben Makhlouf, Borhen Louhichi, Mohamed Ali Mahjoub, Dominique Deneux
(1) Calculate the infinite average plane of the input 3D points. The equation of a plane in 3D space is defined with a normal vector (perpendicular to the plane) and a known point on the plane (the barycenter of the set of 3D points).In order to calculate the normal of the average plane, we should calculate the inertia matrix of the 3D points. This inertia matrix gives the three inertia axes (V1, V2, V3) of the system formed from the set of points. The normal of the plane is the axis V3. The figure (Figure 4) represents examples of the average plane of different sets of points.