10. Planes 2005. 5

Algebraic Definition A plane in space Demonstrative or constructive definition (해석상의 정의) The locus of the points equidistant from two fixed points The resulting plane is the perpendicular bisector of the line joining the two points Quantitative definition (정량적 정의) algebraic definition (Fig. 10.1-3) Ax + By + Cz + D = 0 (A, B, C, D: constant coefficients) Restricted version Ex) C=0  Ax + By + D = 0  a plane perpendicular to the XY plane Ex) A=B=0  Cz + D = 0 a plane perpendicular to the z axis

Two planes have the same direction cosines  parallel Normal Form Plane Defined by Normal Form plane’s normal (perpendicular) distance N from origin The direction cosines(dx, dy, dz) of the line defined by N a = N/dx , b = N/dy , c= N/dz In general, the expressions dx=A, dy=B, dz=C, and N=D are true only if A2 + B2 + C2 =1 z y c b pN O Two planes have the same direction cosines  parallel a x

Plane Defined by Three Points Three noncollinear points in space Implicit equation of a plane Using this equation and the coordinates of the three point Ex)

Vector Equation of a Plane Four way to define a plane using vectors (Section 1.8) a plane through p0 and parallel to two independent vectors s and t Three points p0, p1, and p2 (not collinear) Normal vector  any vector perpendicular to a plane Unit normal vector

Vector Equation of a Plane 3. A plane is by using a point it pass through and the normal vector to the plane The scalar product of two mutually perpendicular vectors is zero 4. variation of the third way Given vector d a point on the plane perpendicular to the plane

Vector Equation of a Plane Normal form of the vector equation (3번째 방법) z y c b pN O a x

Point and Plane Relationships Given point pT, determine on which side of a plane ? Using Reference Point pR If f(xT, yT, zT) = 0, then pT is on the plane If f(xT, yT, zT) > 0 and f(xR, yR, zR) > 0, then pT is on the same side of the plane as pR If f(xT, yT, zT) < 0 and f(xR, yR, zR) < 0, then pT is on the same side of the plane as pR If non of the condition above are ture, then pT is on the opposite side of the plane relative to pR

Plane Intersections The intersection between a line segment and plane Intersection point: pI end points of line segment  (x0, y0, z0) and (x1, y1, z1)

Plane Intersections The intersection between two plane P1 and P2 The geometry of Intersecting planes The plane do not intersect (They are parallel) They intersect in one line The planes are coincident Assigning a value to one of variables (두번째의 경우)