Pocket milling (rectangular) in CNC Programming - Time & Space Complexity

Time Complexity: Pocket milling (rectangular)

O(1)

Understanding Time Complexity

When programming a CNC machine to mill a rectangular pocket, it's important to understand how the time to complete the job grows as the pocket size changes.

We want to know how the number of milling moves increases when the pocket gets bigger.

Scenario Under Consideration

Analyze the time complexity of the following pocket milling code snippet.

G90 G21 ; Absolute positioning, millimeters
G0 X0 Y0 ; Move to start
M98 P100 L10 ; Call subroutine 10 times

O100 SUB
  G1 X[#100 * 10] Y0 F100 ; Mill along X
  G1 X[#100 * 10] Y[#100 * 5] ; Mill along Y
  G1 X0 Y[#100 * 5] ; Mill back along X
  G1 X0 Y0 ; Mill back along Y
M99

This code mills a rectangular pocket by repeating a subroutine that moves the tool along the edges of the rectangle multiple times.

Identify Repeating Operations

Look for repeated actions that take time.

• Primary operation: The subroutine that mills the rectangle edges.
• How many times: The subroutine is called 10 times, repeating the milling passes.

How Execution Grows With Input

The number of moves grows as the size of the pocket increases because the tool must cover more distance.

Input Size (length or width in mm)	Approx. Operations (moves)
10	About 40 moves (4 moves per pass x 10 passes)
100	Still 40 moves, but each move covers more distance
1000	Still 40 moves, with even longer distances per move

Pattern observation: The number of moves stays the same, but the distance covered per move grows with pocket size.

Final Time Complexity

Time Complexity: O(1)

This means the number of milling moves does not increase with pocket size; only the distance per move changes.

Common Mistake

[X] Wrong: "The time to mill grows directly with the pocket size because there are more moves as the pocket gets bigger."

[OK] Correct: The number of moves stays constant; only the length of each move changes. So the count of operations does not grow with size.

Interview Connect

Understanding how CNC programs scale with input sizes helps you write efficient machining code and explain your reasoning clearly in technical discussions.

Self-Check

What if the pocket milling code added a loop to fill the pocket with multiple passes inside the rectangle? How would the time complexity change?

Practice

1. What is the main purpose of pocket milling in CNC programming?

easy

A. To drill holes in a workpiece

B. To cut the outer shape of a part

C. To remove material inside a rectangular area

D. To polish the surface of a workpiece

Solution

1. Step 1: Understand pocket milling definition

   Pocket milling is used to clear out material inside a defined area, usually rectangular.

2. Step 2: Compare with other machining operations

   Drilling makes holes, contouring cuts outer edges, polishing smooths surfaces, so they differ from pocket milling.

3. Final Answer:

   To remove material inside a rectangular area -> Option C

4. Quick Check:

   Pocket milling = remove inside rectangle [OK]

Hint: Pocket milling clears inside areas, not edges or holes [OK]

Common Mistakes:

• Confusing pocket milling with drilling
• Thinking pocket milling cuts outer edges
• Assuming pocket milling polishes surfaces

2. Which G-code command is used to move the tool quickly without cutting during pocket milling?

easy

A. G1

B. G3

C. G2

D. G0

Solution

1. Step 1: Identify G-code commands for movement

   G0 moves the tool rapidly without cutting; G1 moves with controlled feed for cutting.

2. Step 2: Differentiate G2 and G3

   G2 and G3 are used for circular interpolation, not rapid moves.

3. Final Answer:

   G0 -> Option D

4. Quick Check:

   Rapid move = G0 [OK]

Hint: G0 is rapid move; G1 is cutting move [OK]

Common Mistakes:

• Using G1 for rapid moves
• Confusing G2/G3 with rapid moves
• Mixing up G0 and G1 commands

3. Given this CNC code snippet for pocket milling, what is the tool's final position?

G0 Z5
G0 X0 Y0
G1 Z-2 F100
G1 X50 Y0 F200
G1 X50 Y30
G1 X0 Y30
G1 X0 Y0
G0 Z5

medium

A. X0 Y0 Z5

B. X0 Y0 Z-2

C. X50 Y30 Z-2

D. X50 Y30 Z5

Solution

1. Step 1: Trace the tool movements

   The tool starts at Z5 (safe height), moves to X0 Y0, then lowers to Z-2 to cut the pocket edges in sequence.

2. Step 2: Identify final move

   After completing the rectangle, the tool moves back up to Z5 (safe height) at the last position X0 Y0.

3. Final Answer:

   X0 Y0 Z5 -> Option A

4. Quick Check:

   Ends at start point with safe height Z5 [OK]

Hint: Final move is rapid up at start point [OK]

Common Mistakes:

• Ignoring the final Z5 move
• Assuming tool stays lowered after cutting
• Confusing last XY position with intermediate points

4. Identify the error in this pocket milling code snippet:

G0 Z5
G0 X0 Y0
G1 Z-2 F100
G1 X50 Y0 F200
G1 X50 Y30
G1 X0 Y30
G1 X0 Y0
G1 Z5

medium

A. Starting at Z5 instead of Z0

B. Using G1 to move up to Z5 instead of G0

C. Missing feed rate on last move

D. No error, code is correct

Solution

1. Step 1: Check movement commands for safe height

   Raising the tool to safe height (Z5) should use rapid move G0, not cutting move G1.

2. Step 2: Understand feed rate relevance

   Feed rate missing on last move is acceptable for G1 if previous feed rate applies; starting at Z5 is correct.

3. Final Answer:

   Using G1 to move up to Z5 instead of G0 -> Option B

4. Quick Check:

   Safe height moves use G0, not G1 [OK]

Hint: Use G0 for rapid moves up/down [OK]

Common Mistakes:

• Using G1 for rapid moves
• Ignoring feed rate importance
• Confusing start height with error

5. You want to program a rectangular pocket 40mm wide and 25mm tall. The tool diameter is 10mm, and you want to mill in two passes to reach a depth of 4mm. Which approach correctly plans the passes?

hard

A. Cut 2mm depth in first pass, then 2mm in second pass along same path

B. Cut full depth 4mm in one pass along the rectangle edges

C. Cut 1mm depth in first pass, then 3mm in second pass along same path

D. Cut 4mm depth in first pass, then retract and cut again at 2mm

Solution

1. Step 1: Understand depth per pass for tool safety

   Cutting full 4mm depth at once risks tool damage; splitting into equal passes is safer.

2. Step 2: Plan passes to reach total depth

   Two passes of 2mm each sum to 4mm total depth, following same rectangular path for pocket milling.

3. Final Answer:

   Cut 2mm depth in first pass, then 2mm in second pass along same path -> Option A

4. Quick Check:

   Equal depth passes sum to total depth safely [OK]

Hint: Split total depth evenly for multiple passes [OK]

Common Mistakes:

• Cutting full depth in one pass
• Unequal depth passes causing tool stress
• Retracting unnecessarily between passes