Lesson two Views of a Line A spatial line is infinite length, but we usually adopt a definite line segment AB to represent it. Point A and point B are called endpoints. So, the projection of a line is actually the projection of two endpoints on line AB. When Line ∥plane H, the top view, ab, appears as True Length (TS). When Line ⊥ plane H, the top view, a(b), appears as a convergent point. When Line AB is oblique to Plane H, the top view appear as Foreshortened (透视缩短) Line (FL).

1. Three Views of a Line We will get three views of a line if connecting the projections of two endpoints in the same projection plane and brightening them.

In general, a line in space is at angle to three projection planes, respectively. Namely, (Line AB ^ plane H) = , (Line AB ^plane V) =, (Line AB ^ plane W) = .

2. Characteristics of Views of Lines Located in Different Positions in Space According to the spatial position between a line and projection plane, a spatial line may be categorized as: ∥H 1) parallel —— principal lines (投影面平行线) ∥V ∥W ⊥H 2) perpendicular —— lines perpendicular to principal planes ⊥V ⊥W (投影面垂直线) 3) oblique —— oblique line (投影面倾斜线) principal lines and lines perpendicular to principal planes are called peculiar position lines but oblique line is called general position line (投影面的平行线和垂线叫做特殊位置直线而投影面倾斜线叫做一般位置直线).

2.1 Principal lines (投影面平行线) Definition: Line parallel to one principal projection plane and inclined (倾斜) to the other two is called principal line. 2) Kinds: ∥plane H –- Horizontal line (水平线) ∥plane V –- Frontal line (正平线) ∥plane W –- Profile line (侧平线) 3) Characteristics of projection of principal lines

Take Horizontal line for example: ZA = ZB AB∥H : ab= AB (TL) (ab ^ OX) = (AB ^ V) =β (ab ^ OY )= (AB ^ W) =γ H a′b′∥OX ZA = ZB : a″b″∥OY H Frontal line and Profile line are represented in Chart 2.1.

2.2 Lines perpendicular to principal projection planes (投影面垂直线) Definition: Line that is perpendicular to one principal projection plane and parallel to the other two is called lines perpendicular to principle plane 2) Kinks: ⊥H – line perpendicular to Horizontal plane (铅垂线) ⊥V – line perpendicular to Frontal plane (正垂线) ⊥W – line perpendicular to Profile plane(侧垂线) 3) Characteristics of views of lines perpendicular to principle projection plane

Take line perpendicular to plane V example: Front view 1′ (2′) converges to a point. LineⅠⅡ⊥V : Line Ⅱ∥H : 12 = ⅠⅡ (TL) LineⅠⅡ∥W: 1″2″= ⅠⅡ (TL) X =X : 12∥OY Ⅰ Ⅱ H Z = Z : Ⅰ 1″2″ ∥ OYW Ⅱ Line perpendicular to plane H and line perpendicular to plane W are represented in Chart 2.2.

2.3 Oblique line (投影面倾斜线，也称作一般位置直线) 1) Definition: A line oblique to three projection planes is called oblique line. Namely, there is an acute angle between oblique line AB and three projection planes respectively. (AB ^ H) = , (AB ^ V) =, (AB ^ W) = . 2) Projection: ab  AB (SL) a′b′ AB (SL) a″b″ AB (SL) Note: The angle ,  and  are not represented in three views

3. How to judge the spatial position of a line according to the given views ? According to the coordinates, we may judge the spatial position of a line If there is a kind of coordinates value being equal, the line represented by given views must be Principal line. When Z coordinates value are equal, the line must be Horizontal line. When Y coordinates value are equal, the line must be Frontal line. When X coordinates value are equal, the line must be Profile line.

Example1: Judge the spatial position of line according to the given views. Solution: AB ∥ V AB is a Frontal line.

Example 2: Judge the spatial position of line according to the given views. AB ∥ W Solution: AB is a Profile line.

2) If there is two kinds of coordinates value being equal or there is a convergent point (积聚点) in certain view, the line represented by given views must be line perpendicular to principal plane. When X and Y coordinates value equal or there is a convergent point in top view, the line must be line perpendicular to Horizontal line (铅垂线). When X and Z coordinates value equal or there is a convergent point in front view, the line must be line perpendicular to Frontal line (正垂线). When Z and Y coordinates value equal or there is a convergent point in left view, the line must be line perpendicular to Horizontal line (侧垂线).

Example 3: Judge the spatial position of line according to the given views. Solution: AB ⊥ W AB is a line perpendicular to Profile plane.

Judge the spatial position of line AA according to the given views. Example 4: Judge the spatial position of line AA according to the given views. 1 Solution: AA ⊥H 1 AA is a line perpendicular to Horizontal plane. 1

Example 5: Judge the spatial position of line AB according to the given views. Solution: AB ∥ H AB is a Horizontal Line.

3) If there is not coordinates value being equal to views or two views are oblique to projection axes, the line represented by two views must be oblique line.

4. How supplement the third view of a line according to the given views? To supplement the third view of a line is to supplement the third view of two endpoints of a line.

There are two characteristics with respect to points on lines. 5.Points in Line There are two characteristics with respect to points on lines. If a spatial point is on a spatial line, the views of the point appear as on the corresponding view of the line. (从属性) 2) Point dividing a line segment (线段) in a given ratio (比例) will divide any view of the line in the same ratio.(定比性) That is to say: AK: KB = ak : kb = a′k′ : k′b′´ = a″k″ : k″b″

Look at picture please.

Judge whether point C is on the line AB. Example 6: Judge whether point C is on the line AB. Solution: a) Point C locates on the line AB. a) b) Point C dose not locate on the line AB. b)

Example 7: Judge whether point C locates on the line AB. Method 1: Usage the third view. Solution: Supplement left view of line AB and point C. According to the third view, we know point C dose not locate line AB. What is Method 2 ? We may make use of ratio to judge.

Example 8: Given: point K locates on the line AB. Supplement front view of point K. Method 1: Supplement e the third view. Solution: Step 1: Supplement left view of line AB.

Step 2: Supplement left view of point C. Step 3: Supplement front view of point C. What is Method 2 ?

For example 8: Find point C on the line AB to make AC:CB=2:3 Solution: With the assistant of auxiliary line(辅助线) aBo, line AB can be divided into two segments in ratio 2:3. Look at picture please.

In any direction, draw auxiliary Step 1: In any direction, draw auxiliary line through top view a. Step 2: Put 6 equidistant dividing points in the auxiliary line from a to Bo so that we obtain 5 equidistant segments and obtain point Co. Step3: After connecting Bo, b, through point Co draw line parallel to Bob to intersect ab at c. After drawing projection connecting line through top view c, we obtain front view c′. Point C (c, c′ ) is the solution.

6. Relative Position Between Two Lines There are three situations of spatial relationship between two lines which are intersection, parallelism and skew (交叉).

1) Intersecting Lines If two lines intersect then their views intersect and intersecting point conforms to the projection rule of a point.

2) Parallel Lines If two lines are parallel each other then their views are parallel each other respectively.

3) Skew Lines Nonintersecting, nonparallel lines are called skew lines. Skew line locate two planes respectively(异面直线).

There is not a common point (共有点) on skew lines but two pairs coincided points (重影点). PointⅠlocates on line AB and point Ⅱlocates on line CD. PointⅠis above point Ⅱ. In top view, point Ⅰis visible which indicated by 1 but point Ⅱis invisible which indicated by (2).

Point Ⅲ locates on line CD and point Ⅳ on line AB Point Ⅲ locates on line CD and point Ⅳ on line AB. Point Ⅲ is front of point Ⅳ. In front view, point Ⅲ is visible which indicated by 3′ but point Ⅳ is invisible which indicated by (4′ ).

Example 9: Judge whether line AB intersects with line CD. Solution: Line AB does not intersect with line CD. Why ?

Because projection connecting line is not perpendicular to axis OX , there is not a intersecting point (a common point) on line AB and line CD. So, line AB and line CD are skew line but intersecting lines.

Example 10: Judge whether line AB is parallel to line CD. Solution: After supplementing left view of line AB and line CD we know line AB is parallel to line CD.

Example 11: Judge whether line AB is parallel to line CD. Solution: After supplementing left view of line AB and line CD we know line AB is not parallel to line CD.

Example 12: From Point C draw horizontal Line CD to intersect Line AB. Questions: 1) How many solutions ? 2) To draw start from which views ? Who is willing to answer ? Answer 1): There is only one solution in the problem as the highness of the Horizontal line is definite. Answer 2): To draw start from front view. The reason is the same as aforementioned (上述的).

Solution: Step 1: Through c′ draw line parallel to axis OX to intersect a′b′ at k ′and extend to d′. Step 2: Draw projection connecting line k ′k. Step 3: Connect c,k and extend. Lastly, draw projection connecting line d′d. Line CD (c′d′, cd ) is the solution.

Class is over.

7. Theorem of right angle 1) theorem: If AB ⊥ BC and AB ∥ H Then ab ⊥bc. 2) Prove: ∵AB ⊥ BC and Bb ⊥ H ∴AB ⊥ Plane BbcC and AB ⊥ bc ∵AB ∥H, AB ∥ab ∴ ab ⊥ bc 3) The another form of theorem: If AB ⊥ BC and AB ∥ V Then a′ b ′ ⊥b′ c′ .

4) Two views of the theorem of right angle.

Example 13: Given: Line AB and point A Demand: Through Point A construct a frontal line ⊥Line CD.

Solution: Draw frontal line AB ⊥ Line CD. 1) Draw front view: a′ b′ ⊥ c′ d′ 2) Draw top view: ab ∥OX Finish drawing.

Example 14: Given: The edge AB of a rectangle is parallel to H plane. Complete two view of the rectangle. Analyse: ∵ AB ⊥ AC and AB ∥H plane ∴ ab ⊥ac

Drawing: 1) Through top view a, draw line perpendicular to ab. 2) Draw projection connecting Line c′ c. 3) Through point C draw line CD ∥line AB and through point B draw line BD∥ line AC. Finish drawing.