Let's break down the **step-by-step solution** for the union of the three polygons, as outlined in your instructions. We'll go through: 1. **Plotting all polygons based on provided vertices** 2. **Performing the union** 3. **Labeling and symbolizing overlaps** 4. **Creating the attribute table** --- ### **Step 1: Plot the Polygons** **Layer A ("Alpha", value "a"):** - Vertices: (0,5), (4,10), (6,8), (3,5), (6,2), (4,0), (0,5) **Layer B ("Beta", value "b"):** - Vertices: (10,5), (6,10), (4,8), (7,5), (4,2), (6,0), (10,5) **Layer C ("Gamma", value "c"):** - Vertices: (5,8), (2,5), (5,2), (8,5), (5,8) #### **On graph paper:** - Draw each set of points and connect them in order to form polygons. - Make sure to use different line styles for each (e.g., solid for A, dashed for B, dotted for C). --- ### **Step 2: Identify Overlaps and Union Regions** - **Union operation**: Any overlap between polygons will be split into separate regions. - Identify all intersections between the polygons — these are new vertices for the union layer. #### **Types of areas created by union:** - Areas covered by only one polygon (labeled as Alpha, Beta, or Gamma). - Areas where two polygons overlap (Alpha+Beta, Beta+Gamma, Alpha+Gamma). - Areas where all three overlap. --- ### **Step 3: Label and Symbolize Regions** - **Shade**: Where all three overlap (Alpha, Beta, Gamma) - **Diagonal lines**: Alpha & Beta only - **Vertical lines**: Alpha & Gamma only - **Horizontal lines**: Beta & Gamma only --- ### **Step 4: Attribute Table Construction** Each region gets a unique ID. For each region, fill in the values for Alpha, Beta, Gamma as follows: - "a" if the region is in Alpha - "b" if the region is in Beta - "c" if the region is in Gamma - "< null >" otherwise #### **Example Attribute Table:** | ID | Alpha | Beta | Gamma | |----|-------|------|-------| | 1 | a | | | (only Alpha) | 2 | | b | | (only Beta) | 3 | | | c | (only Gamma) | 4 | a | b | | (Alpha & Beta) | 5 | a | | c | (Alpha & Gamma) | 6 | | b | c | (Beta & Gamma) | 7 | a | b | c | (All three overlap) --- ### **Step 5: Final Answer Summary** #### **A) Map Regions (based on intersection):** - Draw union polygons and mark new nodes from intersections. - Label regions by overlap type as per symbolization rules. #### **B) Attribute Table Example:** | ID | Alpha | Beta | Gamma | |----|-------|------|-------| | 1 | a | | | | 2 | | b | | | 3 | | | c | | 4 | a | b | | | 5 | a | | c | | 6 | | b | c | | 7 | a | b | c | --- ### **Key Points for Drawing and Table:** - **Draw each polygon** with its vertices. - **Find all intersections** and mark them as new vertices (diamond markers). - **Label each region** with a unique ID and correct symbolization per overlap. - **Fill in the attribute table** as shown above. --- ### **Explanation** - **Union** splits the polygons into regions based on unique overlap combinations. - **Attribute table** records which input polygon(s) contribute to each region. - **Symbolization** helps visually distinguish types of overlaps. --- If you want an image or a filled-in example map, let me know! Otherwise, above is the complete step-by-step logic and final answer structure for your assignment.

Certainly! Let's walk through a detailed step-by-step process for mapping the union of the three polygons, including calculations, intersections, and attribute assignments. --- ## Step 1: Plotting the Polygons ### **Layer A ("Alpha")** Vertices: - (0,5) - (4,10) - (6,8) - (3,5) - (6,2) - (4,0) - (0,5) (closing point) ### **Layer B ("Beta")** Vertices: - (10,5) - (6,10) - (4,8) - (7,5) - (4,2) - (6,0) - (10,5) (closing point) ### **Layer C ("Gamma")** Vertices: - (5,8) - (2.5,5) - (5.2,2) - (8,5) - (5,8) (closing point) --- ## Step 2: Find Intersections and New Vertices ### **a) Find intersections among polygons** **Between Layer A and B:** - **Edge A1:** (0,5) to (4,10) - **Edge B1:** (10,5) to (6,10) *Check for intersection:* - Equation of A1: from (0,5) to (4,10) - Slope: (10-5)/(4-0) = 5/4 = 1.25 - Equation: y = 1.25x + 5 - Equation of B1: from (10,5) to (6,10) - Slope: (10-5)/(6-10) = 5/(-4) = -1.25 - Equation: y = -1.25x + 15 **Solve for intersection:** Set: \[ 1.25x + 5 = -1.25x + 15 \] \[ 1.25x + 1.25x = 15 - 5 \] \[ 2.5x = 10 \] \[ x = 4 \] Find y: \[ y = 1.25(4) + 5 = 5 + 5 = 10 \] **Intersection point:** (4,10) — which coincides with a vertex of A, so no new point. --- **Between Layer A and C:** - **Edge A2:** (4,10) to (6,8) - Slope: (8-10)/(6-4) = (-2)/2 = -1 - Equation: y = -x + 14 (since passing through (4,10): y = -x + 14) - **Edge C1:** (5,8) to (2.5,5) - Slope: (5-8)/(2.5-5) = (-3)/(-2.5) = 1.2 - Equation: Passes through (5,8): \( y - 8 = 1.2(x - 5) \) \( y = 1.2x - 6 + 8 = 1.2x + 2 \) *Find intersection between these two lines:* Set: \[ -x + 14 = 1.2x + 2 \] \[ -x - 1.2x = 2 - 14 \] \[ -2.2x = -12 \] \[ x = \frac{-12}{-2.2} \approx 5.45 \] Find y: \[ y = -x + 14 = -5.45 + 14 \approx 8.55 \] **Intersection point:** (5.45, 8.55) --- **Between Layer B and C:** - **Edge B2:** (6,10) to (4,8) - Slope: (8-10)/(4-6) = (-2)/(-2) = 1 - Equation: y = x + 4 (since passes through (6,10)) - **Edge C2:** (2.5,5) to (8,5) - Horizontal line: y=5 *Check for intersection with B2:* Set y=5: \[ 5 = x + 4 \] \[ x=1 \] But (1,5) is outside the segment (since B2 spans from x=6 to 4), so no intersection on B2 segment. Next, check B3: (4,8) to (7,5) - Slope: (5-8)/(7-4) = (-3)/3 = -1 - Equation: passing through (4,8): \[ y - 8 = -1(x - 4) \] \[ y = -x + 4 + 8 = -x + 12 \] Set y=5: \[ 5 = -x + 12 \] \[ x=7 \] Point: (7,5), which is on segment B3 (from x=4 to x=7). So the intersection point is **(7,5)**. --- ### **b) Summary of new vertices from intersections:** | Point | Coordinates | Description | |---------|----------------------|----------------------------------| | P1 | (4,10) | Common vertex for A/B (existing) | | P2 | (5.45, 8.55) | A & C intersection | | P3 | (7,5) | B & C intersection | --- ## Step 3: Construct the Union Geometry Using the original vertices and intersection points, the union will consist of multiple polygons created by: - Original polygons - New intersection points (adding vertices) - Splitting polygons at intersection points --- ## Step 4: Assigning Labels and Symbolization - **Unique IDs:** Assign sequential IDs to each new polygon formed after union. - **Attribute assignment:** | ID | Description | Attributes (Alpha, Beta, Gamma) | |-----|------------------------------------------|------------------------------| | 1 | Area only in A | a | | 2 | Area only in B | b | | 3 | Area only in C | c | | 4 | Overlap of A & B | a, b | | 5 | Overlap of A & C | a, c | | 6 | Overlap of B & C | b, c | | 7 | Overlap of all three (A, B, C) | a, b, c | --- ## Step 5: Symbolize Overlap Areas Using the guidelines: - **All three:** shade - **A & B only:** diagonal lines - **A & C only:** vertical lines - **B & C only:** horizontal lines --- ## **Final Summary:** - **Mapped polygons:** Drawn with intersections at (4,10), (5.45,8.55), and (7,5). - **New vertices:** Marked as diamonds at intersection points. - **Union polygons:** Created by splitting original polygons at intersections. - **Labeling:** Each polygon assigned a unique ID and attributes based on membership to A, B, C. - **Attribute table:** | ID | Alpha | Beta | Gamma | |-----|--------|-------|--------| | 1 | a | | | | 2 | | b | | | 3 | | | c | | 4 | a | b | | | 5 | a | | c | | 6 | | b | c | | 7 | a | b | c | --- **Note:** Precise plotting requires drawing the polygons on graph paper or using GIS software for accuracy, but the above steps outline the complete calculation and methodology for mapping the union, identifying all overlaps, and attribute assignment. --- If you'd like, I can also help you with a visual diagram or further detailed calculations for each intersection point!