1. 10. Planes

2. 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 )
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

3. 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

4. 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)

5. 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

6. 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

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

8. 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

9. 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)

10. 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)